There is no authoritative or even moderately reliable way to diagnose "dangerous" mental illness. It is a fairly well established fact that mental illness is more associated with _victimization_ than perpetration of violence. This kind of scaremongering is entirely based in ableism, not actual "common sense."
I consider myself to be more right-leaning and "conservative" but neither I nor any other conservatives I know would support this. Typically, though not always, people on the right want to take away power from the government - not enhance it as this would do.
Nor can you in good faith generalize the ideas of a single person to be representative of all members of the political party that individual claims to be a part of.
I'll add that every conservative in my life would be against this. It actually the kind of thing they hate most about the democratic party policies they complain about: government mandated. Does the topic of vaccine mandates ring a bell?