What I don't understand is, e.g., if $Y_{ist}$ is the employment status of an individual $i$ in a state $s$ at time $t$ after some kind of a policy change $D$, wouldn't there be endogeneity because the employment decisions are based on individuals' choices?
To a degree, but it would depend upon why the policy was introduced. Assignment to treatment at this "higher level" is outside the control of the individual. Some of the randomization is taken care of when the assignment process is not in the hands of the individual participants themselves. On the other hand, endogeneity may be a concern when individual selection into a government training/education program $D_{st}$ is based upon their previous employment earnings.
Why shouldn't (or can't) we estimate the equation with unit, group, and time FEs?
It depends.
Assuming a well-defined level of aggregation and individuals do not move around, then you cannot simultaneously estimate individual, state, and time fixed effects. The individual fixed effects $\alpha_i$ will absorb the state fixed effects $\alpha_s$. The individual fixed effects explain the state level heterogeneity. If I know a person in your sample, then I also know their state membership. Put differently, if I know the individual, then I know whether they belong to a treatment state or a control state. This doesn't necessarily mean you cannot estimate all of these fixed effects in one equation. But, as indicated in the comments, I need to know the specifics of your study. For example, what if people change their residence as a direct result of a state level policy?
In the literature assessing the effects of minimum wage changes on outcomes (e.g., on-the-job training), evaluators working with a panel of individual level data may observe a fraction of their sample move between areas (i.e., states, regions, etc.) over time. Only in this setting would I recommend the estimation individual and areal unit (i.e., state, region, etc.) fixed effects. That being said, it may be worthwhile to work with the "mover" and "non-mover" samples separately.
Same with randomized controlled trial (RCT) evaluations – I often see the inclusion of group FE but so far never of unit or time FE. What would be the reason behind this?
In criminology and public healthy policy circles, difference-in-differences is often used with group (cluster) randomized trials. For example, abandoned housing units $i$ in a major metropolitan city may be randomly allocated to one of several intervention arms (i.e., treatment groups). Once the individual housing units get allocated to each group, evaluators may still use the difference-in-differences estimator with each treatment group in the model. So you may see a series of "group" dummies denoting one or several treatments, not necessarily individual dummies for each housing unit $i$. The model may also include a pre-/post-intervention indicator to identify the before and after periods. That indicator is, technically, a time fixed effect. Using difference-in-differences even after the cluster level randomization is advantageous in the sense that it allows you to distribute all potential sources of bias more evenly.
