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@GerryMyerson You asked why people might be closing. Whether those reasons are valid is a topic for meta. Note that I do not have the ability to close, so I'm only providing a possible explanation. Consider contributing to questions such as meta.mathoverflow.net/questions/1325/… so the community may further refine its requirements.
@GerryMyerson This is a list type question, with no objectively right or wrong answer. As such it could be a list of significant length, and the voting becomes a popularity contest. If the list becomes longer than a page, then you often find better and/or later answers languishing at the bottom while early and/or popular answers sit near the top. See meta.stackexchange.com/questions/124450 and meta.stackexchange.com/questions/238219 and, perhaps most salient, mathoverflow.net/help/dont-ask where this ticks many boxes for questions that shouldn't be asked.
"So I walked into this very innocent-looking combinatorics problem" Sure, sure, that's what they all say buddy. Now face forward towards the camera, please...
Sounds like the real question is, "how do I best teach my students proofs?" To which I'd say, "if proofs are important to learn, they should be used throughout the math program across all math subjects." Whether discussed negatively or positively, proofs are inextricably linked to geometry around here (southeast michigan) and few appear to think of proofs as a separate skill altogether.