I was a crisis intervention volunteer in the early 1970's. We worked though a social service agency serving teenagers and young adults doing outreach for drug overdoses and bad trips, suicides, run-aways, child abuse issues, etc. There was one overdose that was scary. With the rest, I should have been scared but was young enough to think I was invincible.
I've attended to a couple broken shoulders/arms and some cuts, and doled out some moleskin and band-aids.
How did I handle it? If I'm not the victim, there's no need for ME to be upset! Training helps too I think. Keep them from bleeding to death, keep their airway clear, treat for shock, re-assure them, and be the rational one rather than adding to the mess. They would be in a lot worse shape if no one was there.
I've had a stranger give me a ride to help with a broken down car. I've never needed more than that.
"What would you do if you stumbled onto someone seriously injured, and you're many miles from any form of help?"
Stabilize them and go for help. One of my backcountry disaster scenarios concerns that subject. I would leave them with whatever water, food and shelter I could muster and head out with a water bottle and my own essentials in pockets or pack. It is entirely possible for the "rescuer" to become another victim, particularly if you are hiking in the dark to go for help, or you turn an ankle in your haste, etc, so I think it is a good idea to have the classic essentials on your person while going for help. All the classic backcountry horror stories are a cascading chain of screw-ups, so you need to make sure you don't add to the disaster. Be prepared and keep your head.
I don't think there is much more you can do. Of course you would have to evaluate the particular situation, like how busy the trail is, the availability of communications, severity and type of injury, etc.