Ending poverty would certainly help, but I disagree that crime would be fixed. People commit crimes for many reason that aren't related to poverty. Envy, hatred, love, sexual desire, religious fanaticism, political extremism etc. Crimes like murder and rape often have motives completely unrelated to financial status. Not all perpetrators have severe mental disorders either.

In terms of "fixing" people who are violent, I agree in so far that the justice system should focus on rehabilitation and helping people. In many but not all cases, that can be achieved. But generally those people commit crimes first before they're identified. You propose mental health checkups to prevent that in the first place, but many people who are in a bad mental place would not voluntarily go to those. So would you make them mandatory for everyone? That would be quite dystopian, especially with the possibility of being locked up without even having committed a crime. That's exactly the kind of thing I mean by measures that are unacceptably authoritarian. And even then, people would definitely slip through the cracks.