To answer the title question: no, but if it becomes an arms race where it's commonly embraced then God help us all.
As to how it would work: it's different in each case. PSU is using PE as a high-interest loan to finance stadium renovations since they already have $200m in deferred, mandatory safety upgrades that they've been ignoring for the past 1-2 decades while in a weird pissing contest with Oh. St. whereby each of them would go back and forth adding a couple hundred seats to technically be the largest CFB stadium, and I believe largest stadium capacity overall, in the US. UC Los Angeles doesn't have any renovations/upgrades in the work afaik so there's is likely a cash infusion in return for a share of branding and endorsement deals that will leave the Rose Bowl looking like a NASCAR uniform on game days and every break in the action will be filled with PA ads instead of bands playing, etc. - likely with corporate partners of the PE firm. I don't know if PE can outright invest in, and reap profits of, a public school AD but, just playing devil's advocate, if they can there would likely be money upfront and then pressure to cut the fat in the AD such as: marketing and ticket reps as that role has been subsumed by the firm, cutting all personnel deemed non-essential or not profit oriented, cutting of programs that are a net-negative for PE. That last one is a sticking point as it is in violation of Title IX but there are already programs who openly flout it and there would likely be pressure to do so; conversely they could rely on Elevate PE to provide political pressure to challenge Title IX and, in the current climate where we've seen the political power of PE firms, if enough schools follow suit that seems like it's not an outlandish conclusion
