I don't see a similar issue with for-profit schools. In order to attract students they need to to provide good education and foster a good reputation.
Prestige is a funny thing - the fact that a school is hard to get into and expensive may well raise prestige, regardless of educational outcomes.
Don't forget, it's very hard for students and (future) employers/investors/business partners to tell how good an education is. It's quite difficult even for the institutes themselves and any oversight that may exist.
That means incentives aren't as simple as you suggest. Sure, at some very, very long term horizon, better education helps an institute. But that may well be multiple generations away; much too far away to work well in an open market. Especially since there is so much other noise that can be optimized instead.
By constrast, the state's interest in decent education is much more straighforward (not that the politicians making the choices have their incentives straight, but that's another story). In particular, whereas it's clearly in the for-profit institutes interest to milk students no matter how and no matter how unfairly. This is a normal risk in any organisation - but the point is that for-profit organisations aren't magic optimization bullets. These downsides don't need to be a problem, but they do need to be dealt with, and there needs to be some upside to the deal too. What's the upside here?
I think you're clearly right for short-enough courses with clearly demonstrable and measurable outcomes. And I bet you're right in the abstract too, in the sense that "it should be possible to align incentives such that..." - but I'm not sure it holds given the current way education is structured.
Real-world evidence certainly doesn't seem to suggest profit-motive works very well here.
If you're from generations of laborers who have never gone to college, then taking out a predatory loan to go to a predatory school that gives you a somewhat useless degree seems like a good idea at first glance.
Or they just need to dupe people who don't know any better.
A lot of the people who end up at these places are poor, don't have friends or relatives with college degrees, didn't get good advice, and simply don't know how to recognize a scam. Now they have six figures of student loan debt they can't get rid of, and a low-prestige degree that many employers won't even consider.
After all, we have basically that system for for-profit schools and ...
And FWIW, the coding bootcamp I went to (for profit) did more for my career than my non-profit university.