Should you happen to not mount a robust immune response despite being vaccinated then at that point you will have a viral load similar to an unvaccinated person, but the good news is even if this is the case you will still have much less severe disease outcomes than an unvaccinated person.
The vaccines are incredibly effective against all current variants.
It is fair to say that vaccination reduces the chance of infection. "Very rare", however, is not how I would characterize it.
https://www.bloomberg.com/news/articles/2021-07-30/cdc-scale...
Data from Israel suggests that efficacy may be much lower than initially believed, but still much better than nothing.
https://www.sciencemag.org/news/2021/08/grim-warning-israel-...
This is simply not true. In countries with accurate tracing like Singapore and Israel, it's apparent that the vaccinated aren't significantly less likely to get infected than the unvaccinated. Check figure 10 on https://www.moh.gov.sg/news-highlights/details/update-on-loc... ; the unvaccinated only make up around 10-20% of cases. They also only make up 10-20% of the population; if the unvaccinated were more likely to be infected, we'd expect them to constitute a larger proportion of infections.