Of course he should be convicted. If someone is acquitted of a crime and then later boasts about having actually committed the crime, I can not see how anybody could possibly feel that he should not be retried. He even pleaded guilty, it wasn't like there was some new evidence (other than his confession) on which he was convicted.
The fear is obviously that the CPS and police could start hounding defendants with multiple retrials with people being tried over and over again as each new little piece of evidence comes out. That, of course, would be unacceptable. However, this seems unlikely to happen as, to the best of knowledge, the new evidence needs to be extremely compelling in order to take the case to court again (e.g. a confession).
If people start getting convicted at retrial on the basis of tenuous evidence which has been found a decade or two later then I would be uncomfortable. As it is, if someone confesses to a crime and then pleads guilty at retrial, of course they deserve punishment, whether they've managed to fool a jury before or not.