If public higher ed. has its capacity and/or quality cut, wouldn't that increase the demand for private education?

The public universities don't cut capacity or quality - they raise costs. And the private colleges follow suit because they can.

Sure it does, if stake school is X, then private schools need to cost more than X to seem exclusive.

Further, financial aid makes nameplate number less meaningful. If someone has rich parents or wins a full ride then schools want to milk that for as much as possible. Otherwise they can always add discounts down to whatever someone can pay.