If they were mathematicians, they would have published it, no?

Maybe. Writing a publication is a lot more work than a post on 4chan, even if it's a good post.

If it's not your field you might well not bother to write the paper. (After all, it's not your field, somebody might have solved it since it didn't seem that hard, and do you really want to do an exhaustive literature research for a fun puzzle you worked out?)

A chess club I played at was frequented by a guy who used to work as mathematician but got fed up by it at some point and started to work as a chess trainer. Still did maths and blogged regularly but never bothered to publish academic work.

There's a lot of people like that in maths. Very talented but dropping in and out of it or doing it recreationally.

I think Grigori Perelman, who solved the Poincare conjecture and declined the Fields medal and Millennium prize money for it, then said he has retired from professional mathematics to avoid the fame, might qualify as the most famous in recent memory.

Interestingly, his seminal papers on (alongside other major contributions) solving the Poincaré conjecture weren't published in a journal but were simply uploaded on arXiv. Moreover, similar to GP's mathematician may say, Grigori quit professional mathematics not just to avoid fame but also due to disappointment over the field ethics.

Even without considering the reasons already mentioned... isn’t it more fun to be cited as an Anonymous 4chan poster?

Well it was posted on the science and math board so there's a good chance. Headline is a bit clickbaity