>> but sometimes I need you to solve a specific problem for me, so I can keep solving all the other problems I already have on my plate.

Managers definitely need to contribute and this is a great way to do so, and also build credibility with your team. You don't get the fun or deep technical problems; that's not your job, but you can fulfill whatever transparent leadership is (I think?) by protecting your team from the noise (i.e. the shit umbrella) and contributing in a supporting role (the servant part). The hard / tiring thing is doing this consistently and repeatedly.