If you read it carefully, that CDC study compares vaccinated vs unvaccinated in the population of people that recovered from covid.

It does not compare vaccinated (and not recovered) vs recovered (but not vaccinated), but I have seen many people misinterpret it in this way including journalists.

I think this sort of misdirection is by design with how often I see it happening, the right does not have a monopoly on spreading misinformation.