Maybe off-topic because my comments are in the context of Big Sur.
It might not have been just an earlier version of Monterey. I had problems with Time Machine not completing backups after an update in Big Sur until it just started working. Maybe it was related; maybe not. Anyway, I’m glad the backups started completing, for both you and me.
Same here when I upgraded from 11.6.2 to 11.6.3.