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Yesterday I started a discussion related to this on SL, because most of the tsumego there with multiple solutions have been marked as "unsound", which I don't really like.
I think you are right to dislike it. In fact, on the assumption that the problems marked "unsound" were classical problems - which seems likely, given copyright considerations - it is unsound to mark them unsound. It overlooks why the problems were devised in the first place.
I have discussed this at some length in Volume 1 of the Encyclopaedia of Classical Go Problems. The gist of my argument, which is backed with quotations from writers of the time, is that problems then were for recreation with friends. They were topics of conversation. In the absence of television and the like, and even for much of the time a lack of access to books, go problems and new books were seen as objets d'art, to be pored over lovingly and discussed. The more multiple solutions there were, the more to discuss. Almost all problems had allusive names, which provided clues, either for solving or memorisation, but which had intellectual overtones that further stimulated discussion.
This way of viewing problems as part of parties for friends lasted for centuries. In the Ming dynasty, it became more formal with the formation of yuan she, which can be considered a bit more like European salons. Friends were invited as before, discussions took place as before. The new element was some emphasis now on introducing new talent from outside, perhaps with an element of sponsorship. Go masters certainly were part of the new talent invited to the "salons."
Some of these new masters made a name for themselves by pointing out extra lines, or sometimes mistakes, in old problems, and their new variations were typically inserted in new editions. But they never used words like "unsound" or reductio ad absurdum to one line. It was all about adding grist to the mill.
There is one problem I looked at recently (part of Volume 2) which went on for (from memory) 31 moves, whereupon Black was found to be dead (unexpectedly, of course). It was a kaleidoscope of themes. That one problem would have provided entertainment for a whole evening on its own. Some later master pointed out that Black could have made a ko on move 4. True, and most amateurs would have seen that as a possibility anyway. But for Black to play a ko on move could only be justified truly if he had been able to see ahead to move 31 and realised he was actually dead without the ko.
The old collections were never a set of drills to be practised while commuting in your sedan chair on the way to the next court session. The collections were almost totally disorganised. They mixed hard (often VERY hard) and easy, "X to kill" along with "Y to live", and alongside many "what is the status" problems. Life & death were mixed up with tesuji problems and boundary-problems. Also on the mix were pure conveersation pieces such as triple kos. The nearest you get to orderly categorisation is grouping together a few problems which relate to, say, fish or which, for example, all quotations from The Records of the Grand Historian.
To reduce such complex and facinating collections to exercise drills with one solution is pure bowdlerisation.
That is not to say that collections with one-solution problems don't have a place. But they are part of the dumbing-down and fast-food culture of the modern age. Yet, if you judge them by their effectiveness, you could easily argue they don't have a place, really. Almost all modern tsumego collections are the equivalent of learning to dance by watching Avantgardey videos. Those who really want to dance artistically go to schools like the Vaganova Academy, The Royal Ballet School, New York City Ballet, the Paris Opera School, etc. etc. And guess what - all these schools concentrate on
classical technique.