No, the conviction would be overturned and the inmate would be released after seeing a psychiatrist because the guy had to be out of his mind to confess to murder and suffer those unjust consequences. Maybe he would be released to a psych ward before being let loose in public. Maybe not.

no. their sentence would be vacated and they'd be released

A confession alone WILL NOT result in a person going to jail. There has to be additional evidence that the crime to which they confessed actually happened AND they they committed it.

I don't claim to understand the reason, but there are a lot of people out there that confess to crimes they never committed, and even crimes that never happened.

In the scenario you suggest, the person that confessed is unlikely to have gone to jail at all..

If you confessed to murdering and the police had no evidence to substantiate that claim the prosecutor would never charge you without evidence. end of story

That would never happen. A confession alone would not be enough for an arrest. They may detain you, but you would not be charged.

Once the missing person turns up, the person confessing would probably be sent for a mental health evaluation.

Yes because the Judge would assume that the person who confessed was referring to another person that he/she killed.

No - he would be questioned to find out why he lied, but it would seem like he is mentally unstable, or the missing person is lying

that would be up to the judge whether they stayed in jail or not