No, because schools are for people of certain ages.
It's more like having a public senior center that is paid for by taxpayers tell a senior that they can't play with the senior citizen band because they play bingo with the local church (who is the senior groups competition).
I don't think your example is a good parallel. A senior citizen can go to one center, fifteen different ones, or none at all. A child's attendance at school is mandatory in most states, and a child cannot be officially registered in multiple school districts.