not for nothin' and i know it seems like a dodge in modern terms but the church didn't execute people. the civil government did. the church found them guilty of heresy, and the civil government in some cases decided that the appropriate sentence for an unrepentant heretic, who was going to continue to teach others their heresy, was death.
however, i'm not sure where the logic falls apart. there are some crimes that the authority is justified in killing the perpetrator for committing (murder, etc). this is both normal in human experience but also in the divine law given by Christ. this is reaffirmed in the new testament. the governing authority derives its authority from God - Jesus Himself tells this to pilate. this doesn't excuse injustice done by the civil authorities, but the one to hold them accountable for their injustice is God. (this is exactly how ecclesial authority works too, for what its worth).
so, if the state has the divine and moral authority to execute people for crimes against physical personhood, why should they not also have the authority to execute people for crime against souls or corruption of the person? pagans believed this was appropriate. christians believed this is appropriate. only modern secularists believe this is a moral evil - which is weird, because they also don't allow for any absolute morality? the only way you can get there is to assume (probably without meaning to) that physical death is equivalent to the corruption of the person, and being led astray away from spiritual reality to spiritual death isn't real, or isn't worse.
would the state be justified in killing hitler if he openly persisted in his rebellion? does the state reserve the right to execute traitors who are unrepentant and actively working against the stability of the state?
when the state is not secular, and there is a co-equal sphere of religion which both checks the state power, and validates the state power - where the moral agency of the state is derived - an attack on that religion is an attack on the state, no different than treason. and we can see that this is true with how people responded to the reformation - immediate secular strife and bloody warfare, because they 100% understood what this dynamic meant.
why is killing a heretic then a moral question rather than a practical one? i suppose you might disagree with the killing of a traitor if you yourself sympathize with their treason.
is it any more shocking to say "the stat has the right to execute a heretic" than "a criminal" or "a traitor"? or to order a soldier to fight in a war where they may be required to kill? are these unchristian?
the zeal of phinehas was when he ran a spear through an isrealite who was having sex with a pagan priestess-prostitute. God in fact commanded Moses to kill the chieftains of the Israelites who did this. Moses in turn tells the judges of Israel to kill those of their men who had bowed down to a foreign god. during this, while the people were repenting, a person brings one of the foreign women into the camp, and phinehas got the spear. when phinehas did this, he turned back the wrath of God, far from being punished he and his descendants were rewarded with a priesthood. And then God tells Moses to strike the Midianites down.
was God being "unchristlike" there?