What do Army medics do when they reach higher ranks? (Sergeant, Staff Sergeant etc)?

Supervise and teach.

They supervise, if necessary they will step in and do sick call.

lead and supervise younger medics

I was in the Navy but as a CPO I supervised the Jr corpsman, assited in sick call if it was busy or beyond the Jrs experience/training level, I was the liason between the
Navy and the Marine unit to which we were assigned