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I have not really looked at those problems to see if I could find an easy flaw since I first analyzed them, and have been busy with other projects since.It seems a daunting task to figure out if the lack of a solution is due to a plausible oversight of the composer, and what he overlooked. I am not even sure how one would have to approach that problem. The computer just concludes whether there is a mate or not. It doesn't know the concept of 'nearly mate'. Because all the moves are checks it is not uncommon in problems that do have a solution that there is only a single legal move for gote in some positions, or only a single move that does not get him mated very quickly. So having only a single move to escape the mate in some variation doesn't really indicate the composer must have overlooked that move. (Ignoring the problem that the computer usually does not know there is only one refutation to the mate attempt, as it doesn't search any alternative moves once it finds the refutation. This could be solved by altering the program to require at least two refutations in each position, at the expense of driving the solution time up by a factor of 2^N for a mate in N full moves.)

It really requires human judgement to identify a move as a plausible candidate for being overlooked.

Perhaps the following would work: equip the program with a 'conditional exclude-move' option, through which you could tell it: in this position, you must consider that particular move as illegal. You could then start running the engine on the the flawed problem; it will find a principal variation that keeps up the checking as long as possible, but eventually it will run out of checks. You can then try to 'repair' the problem by 'outlawing' the gote moves in this PV one by one, starting at the end, and try to solve the problem for that case. If the outlawed move was the only move that escaped mate, the problem would then have a solotion, and the defensive move could have been the one overlooked by the composer. If the problem is still flawed, because gote has an alternative escape, we probably are already in a position that was never intended to occur.