Is that ethical?

Really depends on the rest: if the justice is fair, if the treatments are really treatments, if only people dangerous for the others are kept, then I think that's probably the best solution we have today.

Ironically, this means that bona fide gangsters will often try to prove they do not have a mental health problem, because ordinary prison at least has a well-defined end date.

I see the irony, but I think it should be the norm. Mental health problems should not be used as a joker to get out of jail. If the system both helps people with a problem and prevents people from using it as an excuse, that sounds perfect.