This is settled science.
Here is a graph of per capita COVID mortality in the US, sorted by vaccination status. Vaccine records are largely unfakeable.
The data is age-adjusted to account for the fact that older people are more likely to get vaccinated. While the data does not account for deaths from the vaccine itself, there has been extensive study of this, and as of 2022 only 55 deaths have been directly attributable to the vaccines, mostly the AstraZeneca one.
The only other potential issue would be a long term side effect from the vaccine that hasn't emerged yet. However, the Pfizer/Moderna vaccines only deliver messenger RNA to cells, and messenger RNA is broken down within a few days, so the vaccine is unlikely to have long term effects, except in terms of increasing your antibodies to the COVID spike proteins.