Are depressed people believed to be master manipulators? Anxious people? Are either of them believed to have no problems with brain function?

I'll give another reason to believe that psychopaths might be better able to help themselves, this time from the summary conclusions:

We believe that the reason for these findings is that psychopaths are fundamentally different from other offenders and that there is nothing “wrong” with them in the manner of a deficit or impairment that therapy can “fix.” Instead, they exhibit an evolutionarily viable life strategy that involves lying, cheating, and manipulating others.

Psychopaths are different in the head. The usual appeals are crafted for the usual heads, by the usual heads.

But I'd refine the summary, noting that while psychopathy may succeed in evolutionary terms, something has not succeeded for their sample of psychopaths because they're in prison, and unlikely to wish to be there.

Has anyone tried to make them better, and more effective psychopaths, psychopaths that wouldn't end up in prison?

I would guess that there are few therapists with a willingness to do that, with the psychological and intellectual capabilities to pull it off. I find the "usual head" quite crazy myself, not very convincing, and likely largely incapable of understanding a paper clip maximizer.