Weighing in with Elendil’s Heir here. The not uncommon view that prosecutors are just rabid wolverines trying to rack up impressive conviction stats is simply false to anyone who knows how the actual system works. The cite he gives represents the law (and actual practice) throughout the common law world. Of course there are exceptions, but they are a tiny minority, like lawyers who steal from trust funds are. Most prosecutors learn that an excess of zeal and rhetorical overreach are counterproductive - jurors are not empty vessels just waiting to be filled with whatever a prosecutor says. They are people who hold at least as wide a variety of opinions as those expressed on this board. And prosecutors know that.
As to the OP, if someone confesses to a crime that manifestly did not occur (such as the assassination of a high profile politician who is still alive) - of course he can’t be convicted. We can all agree on that.
Suppose a conscience torn soul wanders into a police station and confesses to speeding (of which offence there is no LIDAR or other evidence.) In principle such a person might be convicted, particularly if she pleads guilty. But in reality, police would smile, usher her out the door and say with a wink “Well, you got away a with that one. Don’t do it again.”
The most difficult case is that of a confessions to a bodiless murder (by which I mean that class of murder where no body has been found.) The question in the OP is somewhat artificial in that we live in an integrated world where major things like killings rarely occur without jiggling the rest of the social fabric. People are almost always embedded in a social context, even hoboes. If they go missing, someone notices. So if someone comes along and says they killed someone, there will almost always be evidence one way or another that there was a killing, or that there was not. There is rarely an evidential vacuum on that point.
The closest example I can think of is a case where a couple owned and lived on a boat. They befriended a man (the ultimate killer). The accused man was seen on the boat, and the next thing that occurred was that the boat sailed out of the harbour and sailed the South Pacific. The boat’s owners were never seen or heard from again from relatives and others who had a very high expectation of regular contact. The boat was next seen thereafter in the possession of the accused. He gave a series of conflicting accounts as to how he came into possession of the vessel (for present purposes lies like this can be treated as the functional equivalent of a confession.) He was convicted of murder, and the conviction survived appeal to the highest level.
Note the real world evidence that emerged. The deceased were plainly missing. The accused had a motive to profit. He had been seen in their company. Yes there are occasional false confessions by mutters, but for the reasons I have mentioned, they are not difficult to identify. The epistemological boogeyman posited in the OP - that there might be a case of murder where whether or not there was a murder at all is unknown - is improbable.
I have just now thought of another practical example - the burglar’s drive around. Burglar gets caught by a fingerprint and decides to confess because he is stone cold done. Commonly, such burglars agree to do a “drive around” where they take the police around the area and show them the other houses they have broken into. The purpose of this is so that the burglar is sentenced all in one go and doesn’t haveq charges coming in in dribs and drabs messing his sentence and release dates about. Typically police then match the confessions in the drive around up with records of the complaints by the relevant householders, and all is sweet.
But it sometimes happens that when people are broken into, they are unaware of it, and they think they have merely misplaced things or that someone else in the house has picked them up. If the thief just tole some cash lying on a table, the owner might never be able to confirm that. Yet there are convictions in such cases on the basis that the other confessions in the drive around were correct, so the contested one should be treated as reliable.