Compared to life imprisonment it costs the same (or sometimes even more) and has the same outcome of preventing recidivism (re-offending).
The costs of the death penalty are externally elevated. The cost of a bullet is quite cheap.
As far as re-offending-
The murder is not kept in perfect isolation (cruel and unusual), and has the opportunity to re-offend with what are essentially other wards of the state (not to mention prison guards). Anyone who has been around prisons knows there is far more crime in prison than outside.
So what do you do with a person with a life sentence who rapes/kills another prisoner? You have already invoked the worst punishment your scenario allows, and it has failed.
And what of the safety of the other prisoners? Is the state not obliged to keep them safe from further crime? The death penalty ends all future recidivism from this individual permanently.
(It should be noted I generally oppose the death penalty, but as a practical matter understand that it is, and should be, a method of last resort).
Oh, and the conflating with abortion? Pure ideological claptrap.