So the US should stop prosecuting crimes if there is a worry that person might be mentally unstable?

No, but the US absolutely should stop using plea-bargaining as a way to bully people into not going to trial, and then retaliating against people who dare to exercise their right to a trial by adding additional charges against them after plea bargaining. While we are at it, we should reign in the cost of actually exercising the right to a trial -- Aaron lost his fortune defending himself from the government, which is more than most Americans could hope to do.

See, the problem here is that we are talking about giving aid to overly aggressive prosecutors, whose entire careers are devoted to undermining civil rights and destroying their targets. If that is the sort of prosecution you are aiding, yes, you should share in the blame for what happens to the prosecutor's victims.

I think it is fairly uncontroversial to say that the US criminal justice system is a joke. Very few cases go to court (and thus juries) because of the huge amount of power lea bargaining gives to the state.

If you are the aggrieved party, and the punishment the law is meting out is ridiculously out of proportion to the harm done to you, you have some responsibility to speak up.

We already factor mental stability into both sentencing and convictions.

I find that hard to believe. Many prison systems are now the largest providers of mental health services in their various states because so many people with mental illness are imprisoned.