>If I want to film a police officer arresting someone, they can't tell me to stop
to clarify this is legal here as well. Places or people of public interest are excluded from this. So crimes, public figures, monuments, public buildings and so on you can photograph as much as you want. The law protects private individuals, say someone taking your picture specifically on a train, in a gym or what have you.
>You don't own public space
That is true but the idea is that the public space should not become a voyeuristic zoo. Photographing spaces is fine, but individuals should not lose their privacy entirely just because they leave their home.