There is no way that writers get paid every time someone plays their song in public! Not a chance. Who the hell would keep track of it all? You'd have to hire literally millions of people to monitor and report it all and then sort through all the reports, and that's just not feasible. I believe what happens is that samplings are taken on radio and other media outlets of what songs get played during various selected time slots, and from that an approximate estimated total of the number of plays of that song nationwide during say, a month, is calculated, and that determines the royalties.

Can anyone confirm if I'm correct about that?

In the case of the high school marching band, I very much doubt that anyone bothers to catalogue which songs they played last Tuesday and get royalties paid on those.