Doctors, in many respects, are already police. They are among the mandatory reporters with respect to child abuse, for example.
As far as defining the term "mental illness" for legal purposes, well, it's a pretty damn hard term to define for that. It's what I call "kind of a squishy area." The issue of having the doctor's perspective come into play has always been there and, IMO, is impossible to remove.
I'm not saying that I love this bill, and I'd agree that it has some problems. But the problems you describe are in some ways, the same problem with any legislation involving doctors, and no more prevalent in this bill than in any such legislation.
I'm a lawyer, but I'm not your lawyer. If you need some honest-to-goodness legal advice, go buy some.