And absolutely the same logic can and should apply for universities - there can be private exclusive institutions, but the majority should be affordable and mostly paid for by taxes.

This already is the case. 75% of student debt is private universities and colleges.

If you get into Stanford but can't afford it, a loan seems like a good idea, but in reality the loan eligibility should consider the degree and future earning potential (along the same lines of how banks qualify other types of loans).

If we simply cancel student debt or remove private colleges' ability to charge a market rate the result will be no more private colleges (similar to other countries with fully publicly funded education). In these countries you typically have a national exam that determines where you go, or you have to lottery in to a school if it isn't in your district.