It's not just the cost of training. In the US there's implicit expectation that becoming a doctor is a way to become wealthy, similar in earning potential to becoming a lawyer. There's no such expectation in most other countries which are often used in arguments for how socialized medicine "costs less". Their doctors make a fraction of what US doctors do, so of course it "costs less".
Right now in the US you'd be "socializing" the expectation that a doctor should make $300K+ a year, with a very weak upper bound. Which I'm not going to say real good ones shouldn't, but good ones, in my personal experience, are few and far between.
This is something we'd need to deal with before forcing taxpayers to pay for these inflated expectations at gunpoint, _if_ we decide to go the "single payer" route. ACA, ironically, did nothing at all to address the _affordability_ side of things. If I'm forced to give up a good chunk of my income for "free" healthcare, I'd be categorically not interested in paying for yet another beach house some surgeon wants to buy.
Just like all problems involving human motivations, this is a tough one to solve. It's so tough, in fact, that I think the most realistic path forward is gradual automation of everything that can be automated, though technological means, such that we only use doctors where a robot or a computer can't do the job. Someone needs to start another Theranos, but do it for real this time.