Hmm, is that really true though? If so, ideally, people would choose an institute that teaches them as well as possible for as little as possible money. But that's not at all what people do: they prefer prestigious schools that (hopefully) lead to good job prospects.
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.