I'd like to think the Navy wouldn't do this, but they put a NC admiral in charge of Bethesda a while back, with predictable results.
I don't generally object to non-physicians being placed in positions of administrative authority. God knows I'd rather put a gun in my mouth than my name on a door over on the admin wing, and I know I'm not alone on the physician side in feeling this way. Someone's got to do those jobs.
My current dept head is a CRNA, the DSS is a nurse, the hospital XO is a nurse, the hospital CO is a physician who hasn't practiced in forever. Sound like hell? It's not - it all works just fine because they're reasonable people and good administrators who (generally) don't try to micromanage medical care or get in the way of physicians delivering care.
They aren't the ones inflicting online training on me. They aren't the ones who decided I need to spend 4 hours getting mandatory diversity training at the base theater next Monday. The non-physicians in the chain above me are good people. I don't always 100% agree with them, but I'm not deluded enough to think I would if only they were doctors.
All that said, the Surgeon General of the Army should be a doctor.