The initial premise (successful DA realises that Dr McCoy is innocent of murder but about to be executed) is a good hook but even for the fifties the notion that this misstep in an otherwise stellar career, would detail it, seems a little OTT.
History has inferred quite heavily that many judges and 'successful' lawyers were well aware that they were directly responsible for jailing innocent men and women. Some were in the wrong place at the wrong time but more a great number were already in the system and it was just 'their time' to go down for a longer stretch, whether guilty or not.
I would imagine that even in the fifties there would be an acknowledgment amongst jurists that such unfortunate, but wholly necessary, anomalies were part of the model that served the US so well.
The rest of this is actually redolent of Better Call Saul and I didn't expect that.