I understand what you're saying but we'll have to agree to disagree. To emphasize, my point about the third party is simply that the third parties are essentially trying to force all schools to stay because of the fee and because of that dynamic, it might well matter that it's not just two parties. This is something I've seen asserted from a few contract attorneys, so it's not just solely my own opinion.
I've drawn up a few contracts and had them vetted before due to my work in freelance and it's always been explained to me that for a clause to be enforceable, it has to be mutually beneficial to all parties involved. If a clause is being imposed by other signors in order to put one or other parties at a disadvantage, that to me is very relevant.