And your statement that people might have to pay indirectly, I must say I do not know exactly what you are referring to.
Why don't you just ask?
In democratic politics, perception is reality. Most voters believe that their taxes pay their kids' teachers. Maybe I should have inserted "voters believe" in the sentence you quoted. The reference to higher prices is meant to cover my MMT bases. Printing money is inflationary (unless it isn't, which is the point of the article I linked).
You say teachers are underpaid, and I say teachers are underpaid. But they are paid best in capitalist countries. So, maybe the problem lies elsewhere.