https://www.ft.com/content/8af77ab4-e442-11e4-9039-00144feab7de
According to this, even Deanship position is shortage. Provost is higher and much more complicated than Dean.
They kinda do. They ask a short list for applications
Demand supply. 1 President Uni only receives 10 applications.
But one AP position can receive 100 applications.
So President salary shall be at least 10 times higher than AP.I may be wrong but I do not think places like Columbia solicit applications for the position as president.