A better way to compare public and private would be to consider the difference between public roads and toll roads. Some toll roads are owned and operated by state governments, and some by the private sector. But does the driver care who owns the road? I doubt it; the important thing is whether the road is free and open to all, or whether it can be used only by those who can afford to drive on it. The same is true of public and private universities: a university is "public" only if those who need to use it can do so.
In this sense, it seems to me that the malaise that afflicts our public universities is not really about about dollars and cents. If this country can build the world's largest military and fight open-ended wars in multiple theaters across the globe, it can find a way to pay for public education, as it once did, in living memory. But doing so has ceased being a real priority. Affordable "public education" is no longer something we expect, demand or take for granted; to argue that public education should be free makes you sound like an absurd and unrealistic utopian. Meanwhile, we take it for granted that roads should be free to drive on, a toll-road here or there not withstanding. You provide the car and the gas; the state provides the road.
This used to be how we thought about our public universities, before they became exorbitant toll roads: if you had the grades and the ambition, there was a classroom open to you. But if every road was a toll road, no one would expect to drive for free. If every road was a toll road, the very idea that the government would build and maintain a massive system of roads and highways — and then let anyone use it (for free!) — would seem fantastical, ridiculous, even perverse. Anyone expecting the right to drive anywhere they pleased, for free, would be branded a utopian, a socialist, a deluded and soft-hearted liberal demanding a free lunch. That's the world we live in when it comes to highways: when the roads that drive our economy and make modern life possible get too crowded or too congested, we expect the state to build new roads. When the old roads wear out, they are re-paved; when a tree or a landslide obstructs a thoroughfare, the state clears the way. When there are not enough classrooms, on the other hand, the state no longer builds new universities; it simply charges more.