Assume a mass casualty type situation. Something along the lines of AR-558 with physical wounds ranging from phaser and disruptor hits, to concussions, broken bones, and the like.
Each Chief would be from roughly halfway through the run of their series with whatever knowledge and experience they had at that point. Each would have access to whatever kind of portable medical kit a medical officer of their time would have brought for such a situation. There are no able bodied personnel to help. No they don’t have access to a transporter.
Hmm, interesting logic; my first reaction is that even if I program a robot to hit a golf ball I still wouldn’t be any good on the links, but perhaps there’s enough medical theory that she’d have to encode that she would be the top doc. I would have expected the original program to already have the knowledge and skills useful in OP’s scenario, however.
I think all the engineers would have transferable skills, seeing as surgery is basically engineering/plumbing on living things.