In the US it is very easy to get into a university. It is not so easy to get into a good university.
> Why not just build more universities?
Because enrollment and the numbers of students who can pay are dwindling [1].
> Also why not start a private education company that probably wouldn't be eligible to issue government-accredited degrees (who but medical doctors and lawyers needs them anyway?)
The main reason anyone in the US goes to college these days is to get a degree, not to receive education related to any specific job. They want to get a degree because most employers still require a degree either for consideration or for promotion. The employers require this because they see it (fairly or unfairly) as an easy way to weed out those who would be unqualified. Any employer that would accept a non-accredited degree from a private education company that could easily just be selling certificates (which already happens anyway) probably wouldn't even care if you had a degree in the first place.
> yet would provide university-level education quality and compete the free-market way
I just don't see there being a significant market of people willing to pay full tuition for a non-accredited institution, to the extent that it could afford to hire good professors away from existing universities and be a better option for them than pursuing tenured positions elsewhere.
Compounded with the unlikelihood that students would be able to get reasonable loans to attend a school that could very well be a scam, this seems completely untenable. Students get into enough debt as it is going to accredited institutions with government loans.
1. https://www.wsj.com/articles/americas-disappearing-private-c...