# 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?