Because people will still want to go to private universities, same as today.

To the extent that college serves as a signaling mechanism, people may want to go to private colleges even more than today.

Most of the levers available to eliminate public tuition can also be leveraged to restrain private tuition. It's not like expensive private schools don't lean heavily on the feds for funding, both student and research.

Tuition is only part of the costs of going to college. There's room & board, books, various course fees and charges that aren't actual tuition.

Besides private universities, public universities also charge fees for numerous things. In some states where tuition is controlled by the state, this is how universities get around tuition caps when they need more funds than tuition and taxpayer dollars can cover.