I think the data is showing that isn't true, that the numbers are nearly equal in places with a high percentage of the population vaccinated. The only benefit is that the vaccinated don't get as severely ill.
https://www.medpagetoday.com/special-reports/exclusives/9511...
In the UK every age group over 30 is seeing a higher case rate in the vaccinated. I think this is probably explained by a higher prevalence of natural immunity in the unvaccinated cohort but it also shows the vaccines are not providing a high level of prevention of infection: https://eugyppius.substack.com/p/ukhsa-efficacy-stats-death-...
"The evidence is fake, take the mectin".
https://www.factcheck.org/2021/11/scicheck-why-its-easy-to-m...
As far as the cases go the eugyppius post deals with all the supposed caveats and speculations raised in your fact check link. Very funny how they only get raised when the reported case rates are higher in the vaccinated, and not when it was time to kvetch about a "pandemic of the unvaccinated."
It even supports my natural immunity theory at one point:
> Plus, COVID-19 infections among the unvaccinated in the prior four-week reporting period may be “artificially reducing” the case rate for that group in the latest report.
None of these gymnastics would be necessary if the vaccines were genuinely preventing infections at a rate of 6x.