There's much more to higher education costs than Baumol's cost disease. Public universities have much higher tuition because state subsididies to universities have decreased. In 2005, the state paid 3/4 and students paid 1/4 of the cost of instruction at my university. By 2019, those numbers were reversed, with students paying 3/4 of the cost. Total cost of instruction remained the same within a couple percent over that time period.

On the other side, elite private universities are in ever greater demand, driving higher tuitions. Higher tuition is a signal of their elite status, and such universities want to keep their tuition close to that of their competitors. They use financial aid to produce a sliding scale to collect as much money as they can from those who can afford to pay. On the other side, because of that financial aid, raising tuition an amount x may only produce an increase of income of 1/2 x, because the rest goes to increased financial aid. That factor increases the rate at which they raise tuition.

"the alternative is for taxpayers to subsidize it."

You already are subsidising it by providing loans that cannot be paid off before death.

Just need to add debt prisons and we'll be back at 18th century class society.

I think a third real issue is also important, the 'undue hardship' burden necessary to get rid of student loans through bankruptcy needs to be dropped.