Regarding the use of equipment onboard, Starfleet might aim to replace nurses and support staff with machinery so that the ship would have less people to keep alive - or to replace the doctor with machinery so that medicine could be handled by a bunch of generalist mechanics who'd also be doing everything else through expert machines. Both approaches make sense for spaceflight where manpower chiefly means unnecessary expenses.
Starfleet certainly believes in generalists, as evidenced by the background extras wearing multiple "hats" (differently color-coded uniforms) and being seen doing a wide range of things. OTOH, Starfleet doesn't leave specialists ashore: there's a dedicated doctor, a dedicated weapons specialist, a dedicated communicator and so forth. Hoshi Sato would be the first to go in a "realistic" scenario, as she's but translation software on legs and the legs take up unnecessary room... So we can assume either that Starfleet doesn't need to be that stingy and can afford the legs, or that Starfleet can't do the software without the wetware yet. And IMHO we're better off assuming the former. Phlox could be an affordable optional extra as well, then.
As for medical work not being teamwork... Well, it isn't. If a cure pre-exists, a single hero can look it up in the database. If a cure evades our heroes, it will be found by gathering data from the patient and letting the computer mull it over, after which a single hero can look it up. And in Trek, the gathering of data is done by waving a device over the patient or placing him or her on a table - which is inherently far less manpower-intensive than the taking of blood or urine samples (and never mind more demanding tissue samples, perhaps from within) or listening for various vital signs. Really, the medical staff should
consist of a platoon of grunts who simply drag the patients to the machine that does it all, plus the one competent operator of that machine (and her Beta and Gamma shift backups), just like we see!
...Excluding all the old-fashioned surgery needed to reattach severed limbs and close up bodies opened by various calamities. But for some reason (well, an obvious one), we never see those in Trek. Perhaps a patient in need of such surgery is automatically placed inside a proper surgery machine ASAP, as it is understood that no doctor is capable of the speed and precision of a machine in that application?