The infection rate is unreliable because it requires people allowing themselves to be tested when sick, and I would expect people that refuse to be vaccinated also refuse to be tested, on the same grounds.
Your suggested confounder is plausible, but I find the possible confounders in the other direction even more plausible:
- Many places have policies like "proof of vaccination /or/ positive test required", which would inflate the number of confirmed cases in less vaccinated areas.
- Someone with reduced symptoms due to vaccination is less likely to notice them and thus decide to get a test, or to have symptoms severe enough to be hospitalized and thus definitely get a test.
And then the really scary one:
- If vaccines hide symptoms but don't significantly reduce transmission, a vaccinated person is less likely to stay at home during an infection, increasing community transmission.
Further evidence for that hypothesis comes from comparisons of viral load in confirmed cases between vaccinated and unvaccinated people, which finds no significant difference: https://www.medrxiv.org/content/10.1101/2021.07.31.21261387v...