Hello Stephen,
I am glad you liked the post. In this case the ATE is the estimator of the change in the probability of a positive outcome when x2=1 relative to when x2=0 averaged over all observations. The TEM is the same quantity but evaluated at the mean value of the covariates instead of obtaining an average over the observations. In Stata jargon -margins, dydx(1.x2)- and -margins, dydx(1.x2) atmeans.
You pose an interesting question about the LPM. I would need to run some simulations but my preliminary inspection shows that it does not matter for the AME and ATE but makes a difference when you evaluate at specific values of the covariates, TEM and MEM for example. Along the same lines, you might ask what happens when the models are misspecified. I left this out of the post but both logit and probit are equivalently misleading.