I think that public schools and buildings, even off-hours, are another matter because the buildings themselves are intended for gov't activities alone (public education, and so on).
I'm not so sure this intention exists.
In my community, parents take their kids to play at the public elementary school's playground and joggers run laps on the public high school's track. When I was a kid, my AYSO soccer team had practice on the grounds of my public middle school. These are all public, non-government uses of the schools' facilities. It seems to me that off-hours use of a school gymnasium for a Boy Scout meeting would be the same in principle.
Perhaps policies differ from community to community. We live just blocks away from all three public schools - elementary, middle, and high school. In our town, we're not permitted on school grounds at all after hours. Technically, we're not even permitted to park there (though people frequently do at the elementary school when the ball field across the street has no more spaces. The police just look the other way). Town council and the school board does hold meetings at two of the schools after-hours.
And we pay a heck of alot in property taxes, most of which goes to the school district, even though we don't even send our children there.