> My main philosophical gripe is that it doesn't seem to really matter what your major is

That's because (starting in high school) teachers are "assigned" a subject. If they change subjects, they essentially have to restart their career, there is a great personal cost (mostly going back to part-time teaching) for doing that. Plus for a whole bunch of things, from sick days to yard supervision there's a "pecking order". Changing subjects means you go to the back of the pecking order.

So if a biology teacher gets hired to emergency fill a math teacher's position, if they're smart they will never give up that math position.