Is the "constant additive unit causal effect" assumption really needed to interpret a regression coefficient as the ATE? I am reading Unpacking the black box of causality.
At page 768 there is written that, in order to uncover the ATE:

In observational studies, slightly more complex calculations may be needed, although under certain assumptions a regression coefficient can be interpreted as an unbiased estimate of the ATE

and in footnote 8:

Specifically, the assumption is called the constant additive unit
treatment effect in the linear regression

I am looking on the internet for the "constant additive unit treatment effect" assumption, and I have found this slide deck. At slide 11, given the general model:
$$ Y_i (T_i) = \alpha + \beta T_i + \epsilon_i(T_i) $$
The "constant additive unit causal effect" assumption is defined as:
$$ Y_i(1) - Y_i(0) = \beta \;\; \forall i $$
But the ATE is defined as (page 768, footnote 6 of Unpacking the black box of causality):
$$ \text{ATE} = \mathbb{E}(Y_i(1) - Y_i(0)) $$
If we apply this definition to the general model:
$$ \begin{split}
\text{ATE} & = \mathbb{E}\left[Y_i(1) - Y_i(0)\right] \\
& = \mathbb{E}\left[\alpha + \beta + \epsilon_i(1) - \alpha - \epsilon_i(0)\right] \\
& = \mathbb{E}\left[\beta + \epsilon_i(1) - \epsilon_i(0)\right] 
\end{split}$$
and since $\beta$ is a constant:
$$ \text{ATE} = \beta + \mathbb{E}\left[\epsilon_i(1)\right] - \mathbb{E}\left[\epsilon_i(0)\right] $$
so it looks like, in order to interpret $\beta$ as the ATE, all we need is
$$\mathbb{E}[\epsilon_i(T_i)] = 0 \;\; \forall \; T_i=\left\{1,0\right\} $$
we don't need the "constant additive unit causal effect" assumption.
Where am I wrong?
 A: I think the setup is that you want to estimate the average effect of a treatment once in a population where it is constant and once where it is heterogenous. So in both scenarios you want to estimate the same quantity, the ATE, however the "components" of this quantity are different: once constant and once varying across observations. Further, you have a confounder so the treatment is only as good as randomly assigned as given the confounder (this is an observational study).
If you want to estimate the ATE under the constant scenario, you can run a regression of the outcome on the treatment and the confounder.
But if you want to estimate the ATE under the heterogenous scenario you have to take some type of matching estimator. Intuitively, you figure out all the ATEs conditionally on the covariates and then weigh them to get an estimate of the ATE. Now, regression can get you that. But, it is clear that there are some subtleties (relevant in both matching and regression approaches). You definitely need common support that is you need to have both treatment and control observations for each value of the confounder(s).
