I've not seen a convincing argument against simply looking at excess deaths. This avoids all of the problems around classifcations of deaths with/from COVID, vaccination statuses, etc. It's much harder to disguise deaths or lack of deaths. EuroMOMO [1; scroll 60% down to the z-scores] shows excess deaths in Europe basically went back to normal in most countries after the European wave in spring 2020, long before vaccines were available. This was the case despite widely varying levels of lockdown stringency - with at least one country, Sweden, not doing any significant locking down at all. And many countries' peaks at this time were, if you scroll back in time, comparable in magnitude to excess deaths during previous winters. If you draw a line in your mind down the various countries' graphs at early 2021 when significant fractions of the populations of these countries were starting to get vaccinated, you will probably struggle to determine that they've had any real positive effect at all. Once you do that, consider an alternative hypothesis could better describe the data, for example that the virus is highly seasonal and we have very little control over it. Now, a bonus: change the age group from "all ages" to younger cohorts and observe that in many countries excess deaths did not shift from resembling noise during the entirely of the past 2 years. There are even some countries, not known for being paragons of good health such as Scotland, that barely saw any out of the ordinary excess deaths in the young during the whole period. I think people seem to think things are far worse than they actually are in most countries because of the panic created by media and politicians. This is why I am very unconvinced by the argument that vaccines have been helping to end the pandemic, at very least amongst most of those who have been locked down. It seems much more reasonable to me that other factors are at play here.

[1] https://www.euromomo.eu/graphs-and-maps