Yes, some affiliations treat the religion as kind of embarrassing or don’t actually demand anything of their adherents that would be even mildly embarrassing with atheist/agnostic friends. Think Reform Judaism/Unitarian Universalism/ Episcopalianism. Others believe their religion means something.
> The later, impossible for public schools, the former, possibly.
This is actually exactly how the previous dispensation of no religion in public life, judicially started in the US. The Protestants wanted to be able to discriminate against Catholics like you want to be able to discriminate against religions that believe something in tension with the power of the state.