Looking back at Delta/Omicron period (Sep 21 – Feb 22) – that was the longest high-death period in US (taking 1K out of 3K per mln total covid deaths) was the time when boosters started:
US started at significant numbers in October and took it to 30% of population
Israel had 40% of population boosted by Oct 1,
UK started as slow as US but got to 60% of population, others similar or higher.
It is suggestive that slow/lack of boosters contributed to higher mortality during this period, but not a clear proof: UK had peaks before that could have contributed to immunity, some other countries – Poland, Hungary also had peaks (but much shorter ones).