Across US counties, vaccination rates have a statistically significant downward effect on the Covid-19 death rate, as of August 12, 2021. Controlling for poverty rates, age, and temperature lowers the magnitude of the estimate a little. Using the Biden-Trump vote in the 2020 election as an instrument for vaccination rates raises the magnitude of the estimate. Presumably, it corrects for a positive effect of observed local covid deaths on the decision to get vaccinated. Overall, the beneficial effect holds up.
Frankel, Jeffrey A., and Randy Kotti. "The Virus, Vaccination, and Voting." HKS Faculty Research Working Paper Series RWP21-021, August 2021.