Firstly, the melamed is paid for not being able to maintain a full time job because he is a melamed. This is written explicitly in Shulchan Aruch. You are not paying him for the hours he spent learning with your child. You are paying him because in order to teach your child he must forego a parnassa.
Secondly, the reason no one can afford to pay tuition is because tuition is not just money going to teachers. There are a hundred different schools and each one is building a building and paying for other infrastructural costs. Once upon a time a select few learned in yeshivos, of which there were only a few, and other people learned in the city beis midrash/shul or in a small place with a melamed. But now you are paying for the bus to pick up your children, the myriad projects and field trips, the air conditioner and utilities, insurance costs, and everything else.
In any case, Shulchan Aruch says explicitly the community should tax rich people to ensure that the children of people who cannot pay schar limud are educated. The fact that there can be such a thing that a child will not be in school because his parents cannot afford it is against Hashem’s Torah and it doesn’t help that every mosad will wipe their hands because, “so let someone else take him. it isn’t my personal responsibility.” The entire tzibbur is guilty and will be guilty rachmana letzlan when whatever happens happens because a Jewish child was not given a Torah education.