If the state refuses, as a matter of policy, to execute murderers under any circumstances, it rejects the reason people submitted to government in the first place and underlines its own legitimacy. And this isn’t just theoretical bloviation — people sense it in their hearts, even if they don’t think about it in those terms. That was the appeal of Chuck Bronson’s Death Wish movies — when the state fails to carry out its most elementary duty, people will resort to vigilantism, i.e., they seek justice in the only way available to our ancestors in pre-political times.It's true that one of the things that makes a government a government is that it largely has a monopoly on force. But I guess I'm hugely dubious about the idea that governments are made for the express purpose of executing people. And Krikorian's Charles Bronson example is illustrative of that. "Death Wish" came out in 1974—two years after the Supreme Court (temporarily, it turned out) ended the death penalty in the United States. But the crime wave of the 1960s and 1970s had started several years before that. People were already fed up.
I don't think people don't find their government illegitimate when it doesn't execute murderers. But they do find government illegitimate when it can't generally keep a lid on the number of murders, and generally bring murderers to justice. Pile that on top of a whole range of other, mostly lesser crimes, and people don't feel secure in their communities. New York hasn't executed anybody since 1963; the city faced questions of governability during the crime wave—along with a financial crisis—but was reborn in the 1990s thanks to a combination of demographics and policing that had nothing whatsover to do with the death penalty. New York became safer, so people became more confident in the city as a place to work, play, live, and pay taxes.
That's where a government gets its legitimacy: Protecting and serving its citizens. Killing a few of those citizens doesn't necessarily get the job done, especially—as in the case of Troy Davis—when there are real questions of innocence. The State of Georgia in particular, and death penalty jurisprudence in general, face more doubts about their legitimacy today than they did yesterday. It's not because they refused to execute a man.