Advantage of Poisson model over Cox PH model for analyzing first event? For analysis of the first event, I am wondering if Poisson model has any advantage over Cox model, besides some people find rate ratio easier to interpret than hazard ratio?
I notice some people use Poisson model to analyze treatment effect in different strata of time. For example, for COVID studies, they can analyze the effect during a certain variant period. Or they can analyze whether the effect remains the same from 3-6 month compared to the first 3 month. Are these achievable by the Cox model too?

This is pretty much the tradeoff between assuming a particular functional form for a relationship, which can be more powerful if it actually holds, versus working with a semi- or non-parametric model that doesn't depend so strongly on guessing the functional form correctly.
A Poisson model assumes constant hazards over time. It might be fit via the corresponding exponential model for survival times. A Cox model allows the hazard to change arbitrarily over time, only requiring (in its simplest form) that the ratios of hazards associated with covariates be the same over time. If you suspect that hazard ratios change over time, however, a Cox model can be adapted to evaluate that possibility. See the R time dependence survival vignette.
