>A public school in your country is something were any random dude, can just walk in, without anyone being able to object?
These days, generally no, "random dude" cannot just walk into any school as some have the front doors locked during school hours (mostly due to gun violence, not taking photos), but it's not impossible. But "random dude" is not the subject of the article, the subject is student's phones being banned, which does not include cameras specifically being banned, which students could still bring into a school, and it's still perfectly legal to use them. Don't hallucinate something I didn't say to try to make a pointless argument. "Random dude" was not part of this comment thread until you made it up.
>A prison, a nuclear power plant, the hospital and the military are all facilities, that are publicly funded, yet you certainly can't walk in and take random photos or do other things.
You can take photos in the public lobby of all those places, that is accessible to the public, or from outside the facility on public sidewalks or roads, etc. Hospitals over here aren't generally publicly funded, a small number are, but most are for-profit. The US has a lot of problems with healthcare that your country may not have. It's perfectly legal to film military bases from outside the base, anything you can see from a public space is legal to film. It's up to the public entity being filmed to create privacy (walls, etc). Publicly accessible areas are a thing in most public facilities, and restricted areas are a different thing, where filming is not allowed, nor access is given.
1st amendment rights are the same rights that allow security cameras to record people in public, and the public has the right to request police body cam recordings, as well as all camera footage from cameras in public facilities to be requested by the public. If I wanted to, I could get all the security footage from any public facility, just by requesting it. That's why it's also perfectly legal for me to bring my own camera and film inside any publicly accessible area of a public building. There is no reasonable expectation of privacy in public in the US.
>I expect public schools to be publicly funded, not to be the same as any public space. Maybe that indeed doesn't apply in your country, but that sounds bullocks to me.
Constitutional rights don't stop because someone's feelings might get hurt if their photo is taken, and a made-up policy that some institutions like to post on the front door banning photos doesn't supersede the rights given in the constitution. We can ignore those policies, because they are not more important than constitutional rights.
Most people in the US don't even understand 1st amendment rights, but there are a lot of "1st amendment auditors" that go around testing our rights, filming in publicly accessible areas of all the places you've mentioned, and educating people (including law enforcement) that don't have a clue what the 1st amendment guarantees.
You might find some of these videos kind of wild:
Postal Office: https://www.youtube.com/watch?v=uumedKeDqg0
Military Base: https://www.youtube.com/watch?v=ipAN8pft7Fo7
Correctional Facility: https://www.youtube.com/watch?v=egIz6-Ikma4
Filming banks through the window from public: https://youtu.be/9KpUnID-OVc?si=P1OWYxqqPhWnMVVO&t=650