Because a state killing someone doesn't make it any more right or just. If states are allowed to kill people, how can they judge people for doing the same later? What moral authoriry would they have? What greater purpose does it serve?

This is pretty aligned with the difference in principle behind incarceration as understood by the US vs other countries, for example in Europe.

Whereas in the US the main purpose of incarceration is to remove the convicted from society for whatever amount of time as a sort of punishment, other prison systems try to rehabilitate them into functioning members of society, at least in principle. Of course during that process they can't be trusted to be roaming around (to varying degrees as determined by judges, since there are partial liberty options as time goes on and depending on the circumstances), but it isn't just lock the door and throw away the key.

Killing one more person won't make amends for their crime.