The problem is people who actually commit crimes. It doesn’t matter if they’re crazy or not.
The overwhelming majority of crazy people, however you decide to define “crazy,” aren’t a threat to anyone whether they have a gun or not.
So all you’ve got to do is keep guns out of the hands of the people who are going to commit crimes. And defining what is and is not a crime is a lot more clear than defining who is or is not crazy, so it’s much easier.