Competition in the chinuch world is simply not enough to accomplish much. Let’s examine things. To purchase at a store, one can choose to buy it at store A versus store B. Why? Because it is cheaper, or perhaps there is better service. One might make the choice to provide the parnosoh to someone known rather than a stranger. But the product is basically the same. Chinuch bears little to no resemblance to this. Yeshivos are all different. The only common theme is that they house the child for the duration of the school day. The choice element is negligible, as many talmidim do NOT really have much choice where to go, and this refers to parents as well. The school has the option to reject and application, or to keep the talmid out of yeshiva if tuition is not paid. The parent is stuck at that point, and the leverage is such that money comes from somewhere.
The actual competitive issue in chinuch is the mirage that one yeshiva learns more than another, their talmidim get tested by a Rav outside of the yeshiva and do well, or the “name” of the yeshiva is touted as “metzuyan”. These are all fluff, as the production of the talmid who will remain steeped in Torah and Yir’as Shomayim, even if a member of the working world, is still the individual result of a family plus a 1:1 relationship with a rebbe somewhere along the line. It depends on many factors, including one’s individual nature (usually self-motivation), Syatta Dishmaya, and other elements that converge for the single talmid. Our chinuch system contributes quite little to this, and no yeshiva can lay claim to being the producer of the true talmid chochom. They do make such washing statements, but these are empty words. Examine any such talmid, and you will find too many other contributing factors.
No one is really competing for anything today. Yeshivos charge whatever they can get away with, and once they have you, tough. Discussions about midos do not pertain to yeshivos or schools. That issue is for another thread.