You're assuming that mental illness is something that can be cured, and that the cure is not equivalent or worse than killing them.
Quite frankly, our brains are plastic, but not that plastic. It's possible (and we don't know enough about brain structure to say for certain either way) that someone can get into an antisocial enough state that it's impossible, by ethical means, to retrain them, to get them out of that state.
On a motherboard, if you flash your bios wrong you can pull the flash ROM and replace it with a new one. Brains are integrated packages and you can only reprogram them by their IO interfaces, and their IO interfaces are shut down or stuck in a boot loop, or simply running such a hardened update mechanism that any attempt to update the firmware simply adds your attempt to a blacklist and removes the interface you attempted to use.
This analogy is incredibly weak: Human minds are not computers. Put another way: Someone, particularly those on death row, might be so recalcitrant and antisocial that any (ethical) attempt to teach them teaches the opposite reaction, either from spite or from hate. You reward them for good behavior and they make an active attempt not to do that anymore to spite you. You punish them for bad behavior and it only reinforces their hate for you. You do the opposite, rewarding bad behavior and punishing the good, you're making negative progress.
Alternatively, imagine that you have invented a miracle drug that can render a brain fully plastic; Rewrite it to your whim. Is it ever ethical to use such a drug? I say no. It is an unacceptable moral hazard to ever use such a drug. The act of killing is a loss for both parties; Obviously more for the killed, but it is a deep scar that the killer bears, too. Taking on that scar is sometimes worth it. If you were to simply reprogram, from scratch, your convicts into wonderful (or even simply acceptable) people, you would have turned that around. There would be something to gain from "personality death-row". Making it socially acceptable to forcibly reprogram your opponents and feel good about it is a dystopia greater than any that a sci-fi writer has written, and I'm sure many have tried and not gotten the full horrors that it would unleash.