Here are my personal thoughts on the matter.
My vision of this Stack Exchange is, as I pointed out in the answer above, is a place where people can post interesting puzzles and get interesting answers for them, like PCG.SE but for regular puzzles.
To that end, here are some of the puzzle types that I would consider to be unsuitable for this site (and therefore believe should be declared off-topic):
Mass-producible puzzles are puzzles of a specific type that can be produced in large quantities and have a small set of common shared strategies for solving.
I believe that these puzzles are unsuitable simply because there are too many of them, and if too many questions of a single type flood the site, as we saw with the party security questions, the site rapidly becomes uninteresting.
Riddles and language-based puzzles
Riddles, and other puzzles that depend on language (interpreting sentences of words and their phrasing in order to extract meaning from them), have too much inherent potential for ambiguity and inclarity.
This includes lateral thinking puzzles that depends on some quirk in the wording to arrive at the answer. However, it does not include puzzles that make use of just the letters in words in a specific way; I'm specifically talking about morphology and semantics.
This might lead one to believe that only puzzles with a mathematical abstraction are acceptable. I would say it's certainly a good rule of thumb, although a few exceptions might be made.
Of course, questions about the above puzzle types would still be on-topic.
I must emphasize that I'm not excluding the above puzzle types because they're somehow inherently worse. But even on PCG.SE, not all programming puzzles are suitable for Stack Exchange. I had my own experience with that trying to format a king-of-the-hill puzzle on PCG.SE; eventually there were just too many bugs with the format for the verification systems I had in place to be viable. While the question wasn't closed because of it, I would not try and do it again.
Some people believe that this site should strictly be a "Puzzle Theory" Q&A site, in which only such questions about puzzles are asked. If you believe this should be the case, then of course you would believe that none of these puzzles-as-challenges are on-topic, and the question is meaningless. However, I'd have to admit that my own activity would decrease dramatically on such a site, because it would no longer be what I see the site as being.