I wasn’t a CO but I was OIC of a detachment located a few states away from the parent command for a couple of years. I had about 150 Sailors and many needed treatment or hospitalization at various times. The Doc would talk to me about serious stuff but only the bare necessity and only for information. I never had or heard of a Doc or medical letting anyone but the Sailor and the Doc determining what treatment or when for any illness requiring treatment. Only only exception was for elective procedures that would require time off duty. And then it wasn’t how or which elective procedure but just when.
I was also the acting Weapons Department DH on a CVN for about 6 months with about 300 AOs. If you know anything about AOs, if you get hurt, they will. Same thing then wrt medical.
As the Admin Officer in a VP squadron, the Flight Doc technically reported to me (CO wanted this - someone keeping track of him Not normal set up at least back then.). As such, the CO had me gather all the facts and brief him from the Doc. If the Doc came to brief the CO in person, the CO always had me there. Same thing.
I’ve been retired for over 20 years but I can’t imagine this being different now.