If no one else is doing anything about the mess, then it falls to the senior person to sort it out.
As a rule of thumb:
- Ideally your people do the Right Thing by themselves by the magic of 'leadership'. - Second best: you chase the people to do the Right Thing. - Third: you as the senior person do the Right Thing. - Least ideal: no one fixes the mess nor implements the proper way.
I guess some people can achieve the ideal outcome with pure charisma (or fear?) alone, but I find that occasionally getting your hands dirty (option 3) helps earn the respect to make the 'leadership' work. It can also help ground you in the reality of the day to day work.
However, you are right that a senior person shouldn't get bogged down with such work. You need to cut your losses at some point.