Hmm, I don't quite follow your point, honestly. I think: even when treatments begin following hospitalization, Covid-19 still led to that hospitalization to begin with. Doesn't this mean that the injury is still due to Covid-19, not understated or overstated?

And the matched controls in these studies are unvaccinated individuals - either different or the same individuals at different timepoints (i.e., before and after vaccination) - which are compared to vaccinated individuals over a certain period. Similarly, infected are compared with uninfected persons.