Most people don't have to be able to do it, just more than are allowed to do it now, but a factor of say 10.
Furthermore, I don't think people would necessarily need to go through med school and residency.
It's like building a website. There's probably some parallel universe where programming requires certification and only a handful of schools have it. It requires doing MITs SICP as the intro course. Requires compiler construction courses, algorithms, theory of computation and complexity, etc... And then a 4 year apprenticeship with other programmers. And then you can start building webpages, if that's what you want to do.
But we live in a world where it is open and so there's instruction available at many different levels. Would you let someone without a PhD in CS build your website? Of course you would.
And then separately, how many people would trust a self-taught diagnostician?
I trust my car mechanics diagnosis despite the fact I have no idea what level of education he has. But I'm fairly certain he's not an ME from MIT or Berkeley. With good self-study tools, I think that one can self-teach in specialized areas and achieve good mastery rather efficiently. I could imagine Khan Academy for Radiology taught by JHU and Harvard professors to be quite good.