@Nullo I am less concerned about politicians attempting such things than I am private donors. Indeed, the latter have already attempted to wield their influence over schools both public and private (e.g., when Ralph Engelstad bribed the University of North Dakota into ignoring the growing consensus of its students and staff). Politicians, however, are constantly at odds with one another. The message one wants to hear from a university is quite the opposite of the message that another wants to hear. This is an extra layer of protection from abuse, even if it is far from an absolute safeguard.
As for your second concern, I see no better cure for it than an actual education. Prevent people from being as stupid as they would need to be to make such a conflation and the problem is solved. Furthermore, I again would suggest—as I did the first time—that the problem has nothing to do with the source of funding. I went to a private school for part of my undergraduate career, and I met functionally illiterate people there who were admitted for no other reason than their (parents’) ability to pay. I actually have no problem with them being admitted, but I agree that they should not be accommodated.
What sort of institution is more likely to accommodate them, though? In my experience, a private institution. Because I had to pay for college entirely on my own, I began at a community college. The school had placement exams and rigorously enforced standards for students. I knew someone who had to take remedial mathematics because he was a single point below the passing threshold of the exam. When he appealed, he was denied on the grounds that the passing threshold would have been set lower if being off by only one point was sufficient reason to grant an exception. This seems to me precisely the correct decision and response.
When I later transferred to a private school, there were no placement exams. Bad writers were encouraged to go to the writing center to be tutored by English graduate students, but they were not failed for their inability to write a coherent paper. A few of the departments, including the philosophy department, were a bit more rigorous on this score: inability to write was accommodated only at the 100-level or 200-level (courses which can only count as electives and cannot go towards fulfilling major requirements). The justification was that students at those levels may not have yet fulfilled their writing skills requirements. Even that seems too lenient to me, however, as the writing courses were meant to improve a student’s abilities—not to originate them.
My experience with public schools is that they are much more likely to transfer you to where you need to be. Some of my fellow students at the community college were people who had been students at a state university before doing so poorly that they were advised to attend the community college in order to get their basic skills up to an acceptable level (at which point they would be welcomed back to their former school). The university at which I teach has the same sort of arrangement with the local community college. This is the sort of thing that is possible when you have a system of public schools in place (whether it be in addition to or in place of a set of independent private schools).
In any case, I am glad to hear that you are not wholly opposed to the idea. There is no way of doing anything that cannot possibly be abused, so the mere possibility of abuse should never get in the way of a good idea. That said, it is certainly worth considering possible ways in which a system can be abused so as to be on the lookout for both instances of abuse and new methods of preventing abuse. And for my part, I am in no way opposed to the existence of private schools. I think the two types of institution can coexist, and may even complement one another.