I read an article years ago by a priest whose rectory was a block from a state prison. He was called in whenever there was an execution. He said that every man executed there, for decades, died repentant, while most of the prisoners who were NOT facing execution were NOT repentant.
IIRC, the problem of executing the innocent is non-existent.
That may be true. It’s easy to say so when it’s not your life on the line. On the other hand, people under life imprisonment sentences have been released in recent years due to advances in technology or simple reexamination of cases that were once considered “no doubt,” but outlived immediate emotions.