And to expand on this, the data generating process is not about a statistical distribution or any other theoretical construct. Only in the frequentist world do you start with assuming a generating process (for the null hypothesis, specifically).
The data generating process in this case are living, breathing humans doing things humans do.
The potential outcomes are fixed: if a person is assigned to one group the outcome is x1; if another, x2. No assumption is made about these potential outcomes. They are not considered random, unless the Population Average Treatment Effect is being estimated. And even in that case, no distribution is assumed. It certainly is not Gaussian for example.
Under random assignment, the observed treatment effect is unbiased for the Sample Average Treatment Effect. So again, the data generating process of interest to the analyst is random assignment.
And it's my fault for not thinking of that as a possibility. Colour me jaded after experiencing very many bad attempts at randomization that actually suffer from Simpson's paradox in various ways!