Of course I'm not a medical professional, but I'm reasonably decent at statistics. In my view, the statistical correlation across the world to improved results for those vaccinated as well as the decline in the virus when the vaccine started to roll out are statistically a slam dunk. The data is also quite overwhelming as to what can be expected with the various manufactures' vaccine versions.
This virus has more data on it than any other medical device/vaccine/procedure etc. etc. in history and from multiple sources no other medical issue has had as many scientists working on it simultaneously.
In short, one can nitpick here and there about different situations, but the overall picture is as crystal clear as anything is likely to get in the short term. Of note, I've consistently concurred that we do not know if there are any long term side effects and by definition we won't know.