Sure, "all causes" is what we're ultimately interested in. It's just that this detail tells us that something more is going on beyond just vaccination.

Regarding the cohorts - I'm thinking that the explanation for "rates in the vaccinated cohorts start to exceed rates in the unvaccinated" is just that the exceptionally healthy cohort of "early vaccine adopters" is moving onwards to nominally new groups -
- at first its the "Second dose, between 21 days and 6 months ago" which (meaning, in large part the same actual people) then becomes
- "Third dose or booster, less than 21 days ago" by October/November 2021 and eventually
- "Third dose or booster, at least 21 days ago" by December 2021.

After that the gap in non-covid related deaths between the fully vaccinated an unvaccinated cohort starts to close as the fully vaccinated cohort grows ever larger (accepting new and new "less healthy" members it would seem).

The question then becomes - who are the people either getting vaccinated belatedly or stopping to vaccinate and why are they more likely to die compared to either the fully vaccinated or the unvaccinated?

Note that the October-December 2021 period also allows us to rule out the spike in deaths after vaccination hypothesis - while the "early vaccine adopters" are getting their third shot, the mortality in the "Third dose or booster, less than 21 days ago" is very low indeed.