GAW
You have to face the reality of the situation, You can either live in a fantasy world, or you can accept reality.
Do you really think no matter how many times it says dont boot kid, that a Satmar Gvier would give to a Zionistic MO school or Vice Versa
I seriously believe that if the parents were Nisbayish to the point where others knew that people were collecting Tzedaka for them, then we would have significantly fewer non-payers. So did Chazal.
And yes, I do believe that the Satmer (or other Chassidish) will pay for the MO, and vice versa. I have seen it with my own eyes. It does mean though that you need to have tolerance and compromise, which is non-existant in most communities, especially in-town. Out of town, where it is stick together or have nothing, there is more tolerance for other types of frum Yidden.