The death penalty might have deterred crime once, but not anymore. The gap between conviction and execution is too long. Most criminals don’t think they’ll live 20 years anyway, so why worry about the death penalty that will take that long to kill them IF they even get caught in the first place.
But most of your arguments don’t make sense. Again, nobody is saying criminals want to die, so of course when given the choice between a death penalty and something else (something that often means a possible eventual release date) of course they take the something else. That doesn’t mean the death penalty is a meaningful threat, it just means they’re not complete morons. Given the choice between “you’ll be in jail for 20 odd years and we might kill you (assuming it doesn’t get overturned)” and “you’ll be in jail for 12 and we definitely won’t” everybody takes definitely won’t. And both are preferable to shot now.
The argument I made repeatedly: Given a choice of dying or continuing to kill, the perps stopped. Thus, the penalty of death deterred crime. It’s just that simple.