You are right, there's no increased demand. Advertised sticker price and actual price of a private education can be substantially different https://www.wsj.com/articles/private-colleges-see-record-dis...

For decades the private sector has been feeding off the gravitas created by Stanford, Harvard, Yale and other superb private institutions. Maybe time has helped, maybe the Internet made comparisons easier, but it's pretty clear that private schools can be separated into three tiers, with tier one (Stanford, Yale) still being top notch, tier two being a questionable proposition for most people, and tier three (U of Phoenix, Trump U) being borderline fraud.