Paraphrased, your question appears to be:
Why do people posts their math puzzles here, instead of Mathematics Stack Exchange?
To which I'd answer: Because they can.
These questions are on-topic here, as long as they're math puzzles instead of math problems. We like good math puzzles!
- Clever or elegant solution, often an "aha" moment
- Unexpected problem statement.
- Unexpected or counterintuitive result.
Finding a particularly elegant proof is most definitely a puzzle. As for some other reasons besides "because they can":
- Perhaps they are only active here, and prefer to contribute to this community
- Perhaps they want to avoid non-elegant answers by emphasizing that their question is a puzzle
- Perhaps they want some more front-page visibility. Mathematics is an extremely high-volume site, while on Puzzling questions can linger on the front page for days.
To address "there is no puzzle", question by question.
The first one was solved with an elegant proof, as opposed to blind, robotic use of formulas, or standard methods, etc. Figuring out an elegant path to the solution is a puzzle
The second is actually a perfect example of the puzzle vs problem... problem. The first answer presents an elegant proof, doing things that are non-obvious but lead to a nice conclusion. This is solving the question as a puzzle. The other answer throws several formulas together and then brute-forces with computer tools. This is solving the question as a problem. Which answer is more insightful and interesting to read? The puzzle one, of course.
The third explicitly says "Most elegant solution wins!". Again what to do is non-obvious, and again elegance and a puzzle-minded approach is the goal.
I guarantee you that you can keep searching our archives and finding more math puzzles; if they're not "puzzly" enough to count as puzzles vs problems, feel free to flag them for closure.
True, any of these could be posted on Mathematics. Overlap in scope isn't a problem; people can choose either way to go. It seems these users saw their questions as puzzles and not problems and chose to ask here. That's perfectly fine.
Since you keep editing your question, an update from me.
It seems you're offended or upset in some way by people choosing - choosing of their own free will! - to post their mathematical puzzles here.
You bring up mathematical problems which include physics being posted to Physics SE. This is another example of scope overlap. Physics is perfectly fine with high-level mathematical physics problems, or even just regular-level mathematical physics problems. (Relevant meta post) If there's physics in it, it's probably on-topic for Physics SE. Math.SE is one of the biggest sites here - I think only Stack Overflow is bigger - it can survive losing a small handful of physics questions.
You bring up that "classical math methods" are used to solve mathematical puzzles. That's... sort of the point? These aren't lateral-thinking puzzles, the "puzzle" element is figuring out which mathematical methods to apply, and then how to apply them. This is acceptable here.
Scope overlap is fine. When there is scope overlap, it's the user's choice where to post. Which site they choose, and how they choose it, is no-one's business but theirs.