Is there any way we can develop a requirement that every time someone remakes an old movie, they have to go on record saying, "Well X is pretty good, but I think we can make it better?"

No half-assed justifications about making it accessible, or updating it. Just a flat-out claim that they think they can improve the movie.

Or we drop the idea of the "remake" and instead approach this as making a new movie based on the same script. In the theatre, if you put on a play that someone else has done before, you're not remaking it, it's just a new interpretation of the source.

According to a book I read about Roger Corman, they produced three movies from the same screenplay, one time reversing the genders of the protagonist and antagonist. I think it would be interesting if we approached movie remakes more like plays in this way: give the writer full rights over whether more movies are produced from the screenplay. See what happens.