There have been countless times when a Starfleet ship is not in battle and a crewmember has been injured and rendered unconscious. When someone says "Computer, locate X", the computer says where they are. The internal sensors can also read life signs (as evidenced when the computer is queried about the number of life signs aboard). With the speed of these responses, the processes certainly don't take long to complete. Why can't the computer check on each crewmember twice a minute or so to ensure they're safe and conscious, and if they're not, transport them directly to sick bay, or at least alert a medical team?
I believe the answer to this question is the same as for Why can't Star Trek Replicator Technology and Transporter technology be combined to replace vital systems parts to the ship, say a new warp core?
It isn't a question of whether automated systems can do these things but whether should they do them. The amount of autonomy granted to machine intelligences is a choice any sufficiently advanced civilization will have to make and it's a decision that must be weighed carefully. Too much use of advanced technology might eliminate all useful work that some people would want to perform. Systems that eliminate risk might be stifling to people who thrive on putting themselves at risk to test their limits. Too much autonomy might lead to a technological singularity (see The Terminator, The Matrix, Colossus: The Forbin Project, etc.), completely eliminating or enslaving the creators of the machines. The Federation has chosen to build highly sophisticated tools, but to leave their immediate deployment in the hands of people. We can only surmise that they looked at their collective history and decided this was the most prudent course.
I'm not so sure the ship's computer is capable of that.
- Detecting life signs is relatively simple. There's a few basic tests that qualify something as "life", touched on in TNG 1x18, Home Soil.
- Detecting different species requires more work on the computer's part - specifics in their metabolism have to be distinguished, and this is something far easier to do in some cases than others. For example, I believe it was in TNG 7x15, Lower Decks, that one of the junior officers had to help the computer in locating a lost puppy. The computer was incapable of doing so on its own.
- Detecting the difference between normal variations and injuries within a species requires even more intensive scanning, and much more possibility for error - some species have more or less natural variations in their metabolism, that the scans would have to adjust for.
- As for simple physical injury, even that can't always be automated without risk of damaging cultural relations. For example, Klingons have many rituals that involve pain or injury, the first to come to mind being their second Rite of Ascension, which involves hitting the recipient repeatedly with painstiks. Transporting them away in the middle of that would be quite bad.
- Even a simple "life signs are fading" scan wouldn't always be ideal; Klingon culture includes ritual suicide and murder.
These are effectively ships of war. Many scenes feature crewmen continuing to fight or work while injured. A ship could identify that a crewman is injured, but it would be difficult to make the human decision of whether to let a crewman sacrifice himself to save the ship or crew. Should the computer have beamed Spock out when he sacrificed himself to save the Enterprise?
You could blaim the writers. It worked for Mass Effect :p
Despite that Star Trek does try to be a little more realistic with its Science Fiction, some things just must be left to a little imagination and blind faith. Yes, they could wear a device to could monitor as something as basic as vital signs, and assign a computer to give read outs of away team members to the proper medical personnel in the sick bay to expedite the process. But then how would Ensign Ricky die when Kirk needs a bullet catcher?
Also, to expand on your question; no one seems to have invented armor to protect against phasers, lasers, and other cool gagets. I'm sorry, but in a war economy, technology would be rapidly expanding and evolving to deal with threats, and produce new threats. Thus, no drone flyers/bombers, no special forces, strike teams, some sort of laser assault rifle, or the need of carrying more than a remote control gadget that shoots out a cute little beam. Seriously. If they get stuck on a planet (which does happen) what are they suppose to eat? Water might be a concern too. And... unless they're in space, have they ever wore an evironmental/pressure/protection suit of any kind? Are all planets carbon-based 21% oxygen garden worlds with 14.7 psi air pressure? Because that would be awesome.