Chaim, I hear you. I am mostly in the category you described, but I also get occasional discounts when either dealing with a nice principal or with a financial manager who estimates that we have choices (is it gnevas daas to hint that a chassidishe school or a super-modern one is an acceptable option for you :?). As everyone seem to agree, the “full tuition” is set higher than the actual cost.
My point is that the main goal is to create healthy competition, empowering customers (parents), leading to better outcomes. If your system is in the hands of an entity not associated with the school, it might work also. But it will leave parents like me still at the weak negotiating point with the schools.