Strong ignorability: confusion on the relationship between outcomes and treatment In the research area of potential outcomes and individual treatment effect (ITE) estimation, a common assumption called ''strong ignorability'' is often made. Given a graphical model with the following variables: treatment $T=\{0,1\}$ (e.g. giving medication or not), covariates $X$ (e.g. patient history), and outcome $Y$ (e.g. health of a patient). The corresponding visualized graphical model would look like as follows:
$Y \leftarrow X \rightarrow T \rightarrow Y$
(where Y is the same here, see image below)

Then, strong ignorability is defined as :
$(Y_0, Y_1) \perp\!\!\!\perp T \mid X$
where $Y_0 = Y(T=0)$ and $Y_1 = Y(T=1)$.
My question is, if this assumption is made, then this means the outcome is independent of the treatment given $X$. But how can the outcome ever be independent of the treatment? Why do we even bother to solve the ITE problem if we start out with the assumption that the treatment does not really make a difference for the outcome?
Isn't the whole idea of ITE estimation, to determine the effect of a treatment on the outcome Y by estimating the difference between the two potential outcomes $Y(T=0)$ and $Y(T=1)$, one of which we observe as factual observation from our observational dataset?
What I am missing here and why is my understanding incorrect?
I guess it has something to do with the fact, that if we know $X$ (i.e. when X is given), then there is no uncertainty anymore about the treatment $T$ because knowing $X$ makes $T$ deterministic (as we can see from the graphical model above?)
Moreover, I think I do not understand the difference between the following four things:
$Y \perp\!\!\!\perp T \mid X$
$(Y_0, Y_1) ⊥ T \mid X$
$Y_0 \perp\!\!\!\perp T \mid X$
$Y_1 \perp\!\!\!\perp T \mid X$
 A: I'll try to break it down a bit.. I think most of the confusion when studying potential outcomes (ie $Y_0,Y_1$) is to realize that $Y_0,Y_1$ are different than $Y$ without bringing in the covariate $X$. The key is to realize that every individual $i$ has potential outcomes $(Y_{i1},Y_{i0})$, but you only observe $Y_{iT}$ in the data.
Ignorability says
$$(Y_0,Y_1) \perp \!\!\! \perp  T|X$$
which says that conditional on $X$, then the potential outcomes are independent of treatment $T$. It is not saying that $Y$ is independent of $T$. As you point out, that makes no sense. In fact, a classic way to re-write $Y$ is as
$$Y = Y_1T + Y_0(1-T)$$
which tells us that for every individual, we observe $Y_i$ which is either $Y_{i1}$ or $Y_{i0}$ depending on the value of treatment $T_i$. The reason for potential outcomes is that we want to know the effect $Y_{i1} - Y_{i0}$ but only observe one of the two objects for everyone. The question is: what would have $Y_{i0}$ been for the individuals $i$ who have $T_i=1$ (and vice versa)? Ignoring the conditional on $X$ part, the ignorability assumption essentially says that treatment $T$ can certainly affect $Y$ by virtue of $Y$ being equal to $Y_1$ or $Y_0$, but that $T$ is unrelated to the values of $Y_0,Y_1$ themselves.
To motivate this, consider a simple example where we have only two types of people: weak people and strong people. Let treatment $T$ be receiving medication, and $Y$ is health of patient (higher $Y$ means healthier). Strong people are far healthier than weak people. Now suppose that receiving medication makes everyone healthier by a fixed amount.
First case: suppose that only unhealthy people seek out medication. Then those with $T=1$ will be mostly the weak people, since they are the unhealthy people, and those with $T=0$ will be mostly strong people. But then ignorability fails, since the values of $(Y_1,Y_0)$ are related to treatment status $T$: in this case, both $Y_1$ and $Y_0$ will be lower for $T=1$ than for $T=0$ since $T=1$ is filled with mostly weak people and we stated that weak people just are less healthy overall.
Second case: suppose that we randomly assign medication to our pool of strong and weak people. Here, ignorability holds, since $(Y_1,Y_0)$ are independent of treatment status $T$: weak and strong people are equally likely to receive treatment, so the values of $Y_1$ and $Y_0$ are on average the same for $T=0$ and $T=1$. However, since $T$ makes everyone healthier, clearly $Y$ is not independent of $T$.. it has a fixed effect on health in my example!
In other words, ignorability allows that $T$ directly affects whether you receive $Y_1$ or $Y_0$, but treatment status is not related to the these values. In this case, we can figure out what $Y_0$ would have been for those who get treatment by looking at the effect of those who didn't get treatment! We get a treatment effect by comparing those who get treatment to those who don't, but we need a way to make sure that those who get treatment are not fundamentally different from those who don't get treatment, and that's precisely what the ignorability condition assumes.
We can illustrate with two other examples:
A classic case where this holds is in randomized control trials (RCTs) where you randomly assign treatment to individuals. Then clearly those who get treatment may have a different outcome because treatment affects your outcome (unless treatment really has no effect on outcome), but those who get treatment are randomly selected and so treatment receival is independent of potential outcomes, and so you indeed do have that $(Y_0,Y_1) \perp \!\!\! \perp T$. Ignorability assumption holds.
For an example where this fails, consider treatment $T$ be an indicator for finishing high school or not, and let the outcome $Y$ be income in 10 years, and define $(Y_0,Y_1)$ as before. Then $(Y_0,Y_1)$ is not independent of $T$ since presumably the potential outcomes for those with $T=0$ are fundamentally different from those with $T=1$. Maybe people who finish high school have more perseverance than those who don't, or are from wealthier families, and these in turn imply that if we could have observed a world where individuals who finished high school had not finished it, their outcomes would still have been different than the observed pool of individuals who did not finish high school. As such, ignorability assumption likely does not hold: treatment is related to potential outcomes, and in this case, we may expect that $Y_0 | T_i = 1 > Y_0 | T_i = 0$.
The conditioning on $X$ part is simply for cases where ignorability holds conditional on some controls. In your example, it may be that treatment is independent of these potential outcomes only after conditioning on patient history. For an example where this may happen, suppose that individuals with higher patient history $X$ are both sicker and more likely to receive treatment $T$. Then without $X$, we run into the same problem as described as above: the unrealized $Y_0$ for those who receive treatment may be lower than the realized $Y_0$ for those who did not receive treatment because the former are more likely to be unhealthy individuals, and so comparing those with and without treatment will cause issues since we are not comparing the same people. However, if we control for patient history, we can instead assume that conditional on $X$, treatment assignment to individuals is again unrelated to their potential outcomes and so we are good to go again.
Edit
As a final note, based on chat with OP, it may be helpful to relate the potential outcomes framework to the DAG in OP's post (Noah's response covers a similar setting with more formality, so definitely also worth checking that out). In these type of DAGs, we fully model relationships between variables. Forgetting about $X$ for a it, suppose we just have that $T \rightarrow Y$. What does this mean? Well it means that the only effect of T is through $T = 1$ or $T= 0$, and through no other channels, so we immediately have that T affects $Y_1T+ Y_0(1-T)$ only through the value of $T$. You may think "well what if T affects Y through some other channel" but by saying $T \rightarrow Y$, we are saying there are no other channels.
Next, consider your case of $X \rightarrow T \rightarrow Y \leftarrow X$. Here, we have that T directly affects Y, but X also directly affects T and Y.
Why does ignorability fail? Because T can be 1 through the effect of X, which will also affect Y, and so $T = 1$ could affect $Y_0$ and $Y_1$ for the group where $T=1$, and so T affects $Y_1T + Y_0(1-T)$ both through 1. the direct effect of the value of T, but 2. T now also affects $Y_1$ and $Y_0$ through the fact that $X$ affects $Y$ and $T$ at the same time.
A: Doubled has a fantastic answer, but I wanted to follow up with some intuitions that have helped me.
First, think of potential outcomes as pre-treatment covariates. I know this seems like a strange thing to do since the word "outcome" is in their name, but considering it this way clarifies some issues. They represent two combinations of the actual covariates, $X$. So, let's rewrite them as such:
$$Y_0 = f_0(X) \\ Y_1 = f_1(X)$$
(Seeing them this helps divorce them from the observed outcome, $Y$, which we'll get to shortly.) Importantly, if we could observe both of these values, we would not need to assign treatment to anyone. The causal effect of interest is $Y_1 - Y_0$; nowhere in that definition is $T$, the actual treatment assigned, mentioned. This is because we can define the causal effect independently of the actual treatment assignment $T$.
Now, think of $T$, the actual treatment received, as revealing one of the two potential outcomes. The treatment doesn't create the potential outcomes; it merely reveals one of them. The potential outcomes exist in a hidden state prior to treatment receipt, and receiving treatment reveals one of them and leaves the other one hidden. The revealed potential outcome is what we call $Y$, the observed outcome. However, to understand strong ignorability, we don't need to even get to the step where the treatment reveals one of the potential outcomes. Strong ignorability is about potential outcomes (the pretreatment covariates that act as two separate combinations of $X$), not about the observed outcomes. $Y$ does not need to exist (yet) to talk about strong ignorability; it only concerns pre-treatment covariates (including $Y_0$ and $Y_1$) and the mechanism of the assignment of the actual treatment received.
So, prior to one of the potential outcomes being revealed, let's take stock of what we have. We have $X$, the set of pretreatment covariates, $f_0(X)$ and $f_1(X)$, two combinations of $X$, and $T$, the treatment. Unconditional strong ignorability states that $f_0(X)$ and $f_1(X)$ are unrelated to $T$. This would occur if $T$ were randomly assigned or depended only on factors unrelated to $f_0(X)$ and $f_1(X)$. If $T$ depends on $X$, then clearly $f_0(X)$ and $f_1(X)$ are not unrelated to $T$ because both $T$ and $f_0(X)$ and $f_1(X)$ depend on the same variables, namely, $X$.
Conditional strong ignorability (which Rubin calls strong ignorability) simply states that we have observed the set of $X$ that goes into $f_0(X)$, $f_1(X)$, and $T$. Conditional on $X$, $f_0(X)$ and $f_1(X)$ are just constants (potentially plus random noise), and conditional on $X$, $T$ is a random process. It is under this circumstance that we can use specific statistical methods to arrive at a consistent estimate of the causal effect of treatment.

Potential outcomes are confusing. They are typically not taught in an intuitive way, and if they are taught after you have learned about statistics, it is very easy to confuse them with the concepts of the observed treatment $T$ and the observed outcome $Y$, which is what data analysts actually deal with, and the causal effect with a parameter in a regression model rather than as a contrast between two unobserved quantities.
Potential outcomes are abstract quantities that serve primarily as explanatory tools. However, because they are confusing, they are not very good explanatory tools. The graphical (DAG) approach to causal inference is far more intuitive because it relies on the notions of $Y$ and $T$ as they are understood by data analysts. The concept of strong ignorability is isomorphic to d-separation of $T$ and $Y$ in DAG language. Consider reading Pearl's Book of Why to help cement these ideas in an intuitive but still rigorous way.

In response to comments:
The system of structural equations that my description of potential outcomes conforms to is the following (with all variables that are not dependent variables considered exogenous and independent):
\begin{align}
Y_0 &= f_0(X, U_0) \\
Y_1 &= f_1(X, U_1) \\
T &= f_T(X, U_T) \\
Y &= f_Y(Y_0, Y_1, T) = T Y_1 + (1-T)Y_0
\end{align}
This is displayed in the DAG below:

Strong ignorability is that $\{U_{Y0}, U_{Y1}\} \perp U_T$, which is equivalent to d-separation of $T$ and $Y$ given $X$. Note that this DAG is simply a graphical translation of the system of structural equations. There are other ways of displaying potential outcomes in DAGs, one of which is the single-world intervention graph (SWIG).
A: First, you ask a question about $Y_0,Y_1$, but you have a DAG that depends only on $Y$. This may hinder your understanding.
Anyway, simply put, $(Y_0,Y_1) \perp \!\!\! \perp  T|X$ means that there is no hidden confounder, i.e. no factors that both influence the treatment $T$ and the (value of the) outcome $Y$
On a DAG, that simply means that there is no confounder $C$ that points to both $Y$ and $T$. You can have a $C$ that points only to one of them and that's ok.

In order to get a better understanding of conditional probabilities on DAGs, I recommend you to read this (you can see a screenshot of this pdf below).

BONUS (for asking a great question)!
Your initial condition ( $(Y_0,Y_1) \perp \!\!\! \perp  T|X$) plus the condititon of overlap/positivity/common summport ($0 \lt P(T=1 | X) \lt 1$) are called strong ignorability condition. Strong ignorability condition is a sufficient condition for identifying individual treatment effect (ITE). In other words - if no factor both influences taking the treatment and the outcome (no confounding) + all subgroups of the data with different covariates have some probability of receiving any value of treatment (positivity), you can, theoretically, determine the effect of a treatment of any individual from your data.
Edit: if you want to draw a DAG with $Y_0, Y_1$ instead of $Y$, it should look like below (the yellow part is the confounder part)

