Domain management, treasury division, training, spell research, item/scroll/potion making, business interests, investments - none of these involve adventuring*. All are things done while out of the field, and thus fall under downtime.
Another major and fairly frequent downtime activity at the table, not yet really touched upon, is levelling up.
* - usually; and if-when they ever do the other three pillars/modes boot up.
Agreed.
Seeking immortality is most likely either itself an adventuring activity or will involve a lot of such.
I love shenanigans as much as anyone but I don't see it as its own pillar as there is no place or time where it's not overlapping at least one other pillar (if downtime is counted as a pillar). That said, I think it could almost count as its own mode of play under
@Charlaquin 's model.