Check out how it works in other countries that have both publicly funded and private education.
The losers here would not be the elite private institutions, which will always have high demand and did even 50 years ago when public higher education was sufficiently subsidized that almost anyone could afford it with a summer job.
(Many of those institutions are already offering free tuition to lower and middle income students anyway. And they’re all incorporated as nonprofit entities.)
The losers would be the low quality for-profit degree mills that advertise on the subway and radio and charge students ridiculous tuition for degrees that won’t help them get ahead.