However, execution lets the convicted person off the hook the easy way compared to a lifetime of incarceration.
That's irrelevant, as the justice system is not to be a method for taking revenge, but to make society a better place to live in, with less crime.
The death sentence is flawed for other reasons. Almost all murders happen either in affect, or in a situation where the perpetrator thinks he can get away with it. In either case, having the death penalty will have no effect on whether a murder will happen or not. And it might lead to more murders, because if there's a death penalty in place, the perpetrator has nothing to lose by killing witnesses, cops, or anyone else who might get them arrested, now or in the future. The rational decision for them is to do anything not to get caught, including more murders.
Also, the costs of a death row inmate by far exceeds the costs of a long term imprisonment. (This is particularly true in the states that allow prison slave labor - which has a high correlation to the states that allow capital punishment). The many rounds of appeals that a death sentence automatically trigger cost a heck of a lot more than the room and board.
Then there are the cases of people who have been wrongly executed. One case is one too many. And a peer-reviewed study shows that as many as 4% of people convicted to die are likely innocent.
Unless there's a way to bring people back to life again, that in itself should be enough to put a stop to it.
But the unwashed masses want panem et circenses, and revenge, not justice. So the show goes on. And innocent people die.