in chapter 9 of gelman's data analysis using regression and multilevel/hierarchical models, page 170 presents a simple example on the bias of an omitted variable $x$ from a regression of an outcome $y$ on treatment $T$. It uses three regressions:
correct specification including $x$: $$y_i = \beta_0 + \beta_1 T_i + \beta_2 x_i + \epsilon_i$$
omitting the confounding variable:
$$y_i = \beta_0^* + \beta_1^* T_i + \epsilon_i$$
and regressing the confounding variable on the treatment:
$$x_i = \gamma_0 + \gamma_1 T_i + \nu_i$$
then it substitutes this last expression into the first regression to arrive at
$$y_i = (\beta_0 + \beta_2 \gamma_0) + (\beta_1 + \beta_2 \gamma_1) T_i + (\epsilon_i + \beta_2 \nu_i)$$
and equates $\beta_1^*$ to $\beta_1 + \beta_2^* \gamma_1$ to show that when there is association between the omitted variable and the treatment or between the omitted variable and the outcome then neither $\gamma_1$ or $\beta_2$ are, correspondingly, zero, so there is bias in the estimate of the effect of $T$.
Later, on page 188 the author advises against "controlling for variables measured after treatment / mediators / intermediate outcomes" by using an example of a home visits treatment $T$, outcome $y$ corresponding to a cognitive score on children, and parenting quality $z$.
The author argues that including $z$ in the regression of $y$ on $T$ is problematic because there is an effect of $T$ on $z$. So the coefficient of $T$ in such a regression corresponds to a comparison of units which are identical in $x$ and $z$ but differ in $T$ - but because $T$ has an effect on $z$ these units automatically have different $z$ characteristics (if $T$ is positive, then the unit that did not receive the treatment has lower underlying $z$ "parental skill" than the other in order for the two observations to have the same observed value of $z$). While this makes sense, I am having trouble reconciling this with the earlier example of omitted variable bias. Can't the same be said in the earlier example when $x$ is not omitted? i.e. including it when $T$ has an effect on it implies comparing units with different $T$ values, fixed $x$, but thus different underlying "$x$" characteristics? How are the two cases "different"?