Even more reason for the schools to not get involved. They would have to track how many permits have been issued, how many are left, and how many of those they want to use themselves vs leave for other schools, etc.
And what happens when some students don’t get their visa? Will the government provide updates on who was approved and who was denied? Are the schools then going back through their waiting list to admit others in place of those denied? And what happens if some of those are also denied?
It gets very complicated very fast. My bet is on the schools letting the visa process play out independently, and not get involved.