There is also the money. Sure you can train 10-100x physicians, but total compensation won't grow as fast, and so you should see lower overall wages (supply and demand). But it's still more complicated than that, because some people choose not to undergo X or Y procedure for various reasons, insurers don't cover or whatever. So even if you could train enough doctors to fulfil the whole population's needs, that doesn't mean you will have: 1. enough equipment, material, operating rooms, etc to do the procedures 2. enough supporting staff, and on and on.
The most common reason I hear from MDs against training much more doctors is the first one I gave you: less practice means a more dangerous practice.
My own opinion is relatively simple. Medical schools should stop screening applicants as much and let students join freely.
Medical schools are all about getting well-rounded individuals who did extra-curriculars etc. They usually view it as a good thing that you have life experiences outside of medicine, like working or studying an entirely different subject. However it works one-way: you can't (generally) just go to med school, learn the subject matter and then move on to another field, enriching _that_ one with your experience. I have come to believe that if we are to see any improvement, med school will have to open up just like every other discipline. The current system of stressing out students for a few years before even being allowed in the classroom is, IMO, partly responsible for the job dissatisfaction that you see.
Now freeing up med school admissions does not mean allowing everybody to _practice_ medicine. Prospective students would actually get a chance to learn the material for a few years before having to interview for internships. So instead of filtering _before_ med school and forcing a huge sunk cost psychological barrier to students, you can let the students figure out by themselves if they actually enjoy the subject matter. It's not a perfect solution but I am quite certain that society as a whole would benefit. It is entirely unfair that medical knowledge (not practice) is restricted to a lucky few.