Maybe I'm seeing this wrong but a public school is a government institution, right? It seems reasonable, then, that the government could say "no phones in here", just like they can say "no drinks in the courtroom".

Not all schools are government institutions or funded by the government. But it wouldn't matter to your point, because by that logic anyone who owns the building should then be able to make rules about what goes on inside that building, or any organization, etc. Much like how you have basic wardrobe etiquette, you could also have a rule that says no phones in a classroom.

Now me being Estonian, just across the sea from Finland, I was surprised to learn this needed to be a law in the first place since when I went to elementary/primary school smartphones and laptops were not permitted in the classroom. Didn't need a special law for that, it was just the school's rules. You either play ball or you get called into the principals office. This happens enough times and you simply get kicked out of the school.

Well sure, I tend to agree with that. Government buildings can be an exception because they're public and we more carefully constraint the things the government can do, and because you can be forced to go there.

I too am a bit baffled by the idea that you need a law for that. But, as I said, we have a different common law tradition in America that changes things. The only two big legal influences which I've read on are common law and the justinian codex -> napoleonic code path; scandi law is alien to me.