I think the NHS is overburdened partly because doctors are also having to pick up on social problems which at one time would have been dealt with by family or local community groups such as church ministers. The care of elderly people, for instance, is now a medical issue instead of a social one. We're never going to go back to the women of the house also being the carers of the elderly so more money needs to be put into the system to provide a decent level of care.
I do think people rush to the doctor too often for minor problems. I can't be bothered, myself, because it's so much kerfuffle to get an appointment. As my mother's friend used to say, it's pointless making an appointment, I'll either be better or dead by the time it arrives!
Regarding a shorter training for doctors; my daughter is in her 3rd year of training and she hasn't even covered all of the body systems yet, so I don't think that will work.
As for the US system, my son and family have good insurance. Even so, when their 6yo stuck a piece of Lego up his nose, they had to pay $900 on top of what their insurance paid out! The mind boggles as to how much the total came to.