WIY why make it the school’s responsibility? Besides many bais yaakovs make plays for precisely that reason. They have practices scheduled for most of the long motzei shabbosim in order that the kids have a fun option so they’ll stay away from the hang outs.
A schools responsibility is chinuch, that’s it. It’s to teach what the Torah says in a way that sounds appealing. Also being that the students are in school for large portions of the day the school should be looking out for the emotional well being of the students. But that’s during school hours. They’re not a babysitting service like many parents seem to think.
That being said if parents don’t want their kids to do xyz they should be proactive in preventing it. Merely forbidding something won’t result in any change. The parents gave birth to their children so its their responsibility. If a parent doesn’t want to entertain their kids they can find something for their kids to do, like a chug or something like that. When I was younger my parents would take us on drives
Or at least have something interesting for us to do on motzei shabbos so we weren’t roaming the streets.