> but ultimately the virus spreads no matter what.
No, the virus spreads as a function of what actions you take and society slows based on what actions you take. If no one is dieing or going to hospital you choose economy and cases go up but if you are highly vaccinated then the consequences are not enough to not open up further and you run with higher numbers of known cases.
A correlation study is irrelevant to how effective vaccination is, just as any other intervention might appear ineffective if it allows systems to remove stricter restrictions with higher societal impact or makes people feel safe enough to participate in additional activities.
Some unvaccinated people in my town feel safe enough to do things because more people are safe and are doing them than in a town with no vaccination and more conservative average behavior, that doesn't mean the vaccine didn't work.