How would you describe the uncoundedness/ignorability assumption to
somebody who has not studied the RCM?
Regarding intuition to somebody not versed in causal inference, I think this is where you could use graphs. They are intuitive in the sense that they visually show "flow" and they will also make clear what ignorability substantively means in the real world.
Conditional ignorability is equivalent to claiming $X$ satisfies the backdoor-criterion. So, in intuitive terms, you can say to the person that the covariates you chose for $X$ "blocks" the effect of common causes of $T$ and $Y$ (and do not open any other spurious associations).
If the only conceivable confounding variables of your problem are the variables on $X$ itself, then this is trivial to explain. You just say that since $X$ contais all the common causes of both $T$ and $Y$, that's all you need to control for. So you could say to her that's how you see the world:
The more interesting case is when there might be other plausible confounders out there. To be more specific, you could even ask the person to name a potential confounder of your problem -- that is, ask her to name something that causes both $T$ and $Y$, but it's not in $X$.
Say the person names a variable $Z$. Then you can say to that person that what your conditional ignorability assumption effectively means is that you think $X$ will "block" the effect of $Z$ on $T$ and/or $Y$.
And you should give her a substantive reason why you think that's true. There are many graphs that could represent that, but say you come up with this explanation: "$Z$ will not bias the results because even though $Z$ causes $T$ and $Y$, its effect on $T$ goes only through $X$, which we are controlling for". And then show this graph:
And you could think of other cofounders and show to her how $X$ is blocking them visually on the graphs.
Now answering the conceptual questions:
Specifically, if T is the treatment, shouldn't the potential outcome
be very dependent on it? As well, if we have a randomized controlled
trial, then automatically, . Why does this hold true?
No. Think of $T$ as the treatment assignment. What it says is that you are assigning the treatment to people "ignoring" how they respond to the treatment (the counterfactual potential outcomes). A simple violation of this would be you tending to give the treatment to those who would potentially benefit the most from it.
That's also why this automatically holds when you randomize. If you pick the treated at random, this means you did not check their potential responses to the treatment to select them.
To complement the answer, it's worth noticing that understanding ignorability without talking about the causal process, that is, without invoking structural equations/graphical models is really hard. Most of the time you see researchers appealing to the idea of "the treatment was as-if random" but without justifying why that is or why that's plausible using real world mechanisms and processes.
In fact, many researchers simply assume ignorability for convenience, in order to justify the use of statistical methods. This passage from Joffe, Yang and Feldman paper speaks an inconvenient truth most people know but do not say during conference presentations: "Ignorability assumptions are usually made because they justify the use of available statistical methods, and not because the are truly believed."
But, as I have said in the beginning of the the answer, you can use graphs to argue about whether a treatment assignment is ignorable or not. While the concept of ignorability itself is hard to grasp, because it states judgements about counterfactual quantities, in the graphs you are basically making qualitative statements about causal processes (this variable causes that variable etc), which are easy to explain and visually appealing.
As mentioned in a previous answer, there's a formal equivalency between graphs and potential outcomes. Hence, you can read potential outcomes from graphs too. Making this connection more formal (for more see Pearl's Causality, p.343), you could resort to the following definition: the potential outcomes would stand for the total of all variables (observed and error terms) that affect Y when T is held constant.
Then it's easy to see why ignorability holds in RCT, but more importantly, it also allows you to easily spot situations where ignorability would not hold. For example, in the graph $T \rightarrow X \rightarrow Y$, T is ignorable , but T is not conditionally ignorable given X, because once you condition on X, you open a colliding path from the error term of X to T.
To sum up, many researchers make the ignorability assumption by default, for convenience. It's a convenient way to assume the sufficiency of a set of controls without needing to formally justify why that's the case, but to explain what it means in a real context for a layman, you would need to invoke a causal story, that is causal assumptions, and you can formally tell that story with the help of causal graphs.