You just described supply and demand.

If the wages are too low, you’ll have a shortage.

To remediate the shortage, wages must increase.

Yeah but step 2 of that is "wage increase" (then step 3 of that is "either school A increases wages or school B increases wages and gets all the teachers, etc.) You can't arbitrarily pick a point at which you stop making an appeal to authority to Econ 101; it's all or nothing.

The point here is, the situation should make us ask ourselves "huh, I wonder why wages aren't increasing." But I can save you some time and tell you the answer is mostly we don't think those jobs are high status enough.