I watch a lot of US fictional programmes

This scenario has happened a few times, mainly on cop shows, but once on a hospital based one.

A guy is on death row, and due to be executed in a few days.
He is then found collapsed for one reason or another, and the warders are falling over themselves to administer CPR etc.

In the hospital one, the guy suffered heart failure, was rushed to the hospital, and they were considering operating and giving him a heart transplant.

Does this happen for real.  If so, why?
What is the point. 
Surely a waste of resourses and doctors time which could be better spent treating more worthy cases.