It's not an assumption but rather an observation looking at the data - there are just less deaths in this group. It does not matter here that the older were a majority among the early vaccinated since we are looking at age-weighted data anyway.

Well, "ruled out" in the sense that if the shot itself would have been a major cause of death then it would have to show up in that Oct-Dec 2021 data as well. It's just a counterpoint to that idea. If you are interested in shot 1, then look at Jan-Mar 2021 when the first vaccines were rolled out. This is the same "healthy" cohort, and the non-covid mortality rate there is half of what it is for the unvaccinated.