We have such public colleges, of course. Not the kind mentioned around here much, but the kind that are serving many, many families.
But there are other families willing and able to pay more to get something different.
And it is theoretically possible they could only do so at private colleges. Then all the families willing and able to pay more would go private, and the only students at publics would be from families unwilling or unable to pay more.
But, long ago, the whole “land grant” movement was based on the premise that was not a good outcome, that at least some public colleges could and should compete to provide the sorts of colleges that these families wanted. And the end result was an extremely robust post-secondary education system.
Which is not to say that proves it was a good idea, either in general or in all details. But I do think it is worth noting that unless we banned private colleges, then stopping public colleges from trying to compete with private colleges would very likely result in a much starker difference between the sorts of colleges people attended depending on their families’ respective willingness and ability to pay.