Yes.

It does make sense. Police are absolutely not beyond reproach, and there's screwups all the time. They need to be held to a high standard.

It's also easy to imagine reasonable compromises, like a time delay where they only have to report after e.g. 48 hours, and allowing a system whereby a judge can issue a warrant to extend that delay.

That's more or less what a search warrant is, so yes.

Specifically, I believe that if information that is held by a private 3rd party I accessed by anyone, law enforcement or otherwise, that third party should be required to tell you that it was requested and by whom. Just like they can't put a gag order on a search warrant to your home, this hypothetical search would be exempt from gag orders.