> The person allocating the money says "I'll give you {amount}". The person running the education system says "{amount} will let us educate {number}".

Nah, they say "I'll give you {amount} to educate {number}", and then the educator tries to do as much as possible with the resources they got. There is no negotiation about either {amount} or {number}, other than trying to argue that amount or number needs to be changed because quality or industry wants more/less.

My point stands - the number who "qualify" is decided by two bureaucracies. One decides how much money to spend, the other spends it.

In such a system, it's typical that one bureaucracy decides what "qualify" means.

And another decides what "educate" means.