You mentioned chess, can I know which betting site offer chess? it should be accept cryptocurrency because we're on Bitcoin forum. I mean betting on chess where the bettor is the one who play the game, not betting on someone who play on a tournament.
If someone wants to train their mind, they can still go for other mind games that doesn't actually require them to gambling with real money, some people can feel convinced that gambling can train their mind but once they dive in, it becomes a different story entirely. I don't disagree with you that gambling is a mental sport but in the way you described it, it's result will be different for every individual, it can only be used like that and would prove positive result in some individuals but for many people, you will get an opposite result since they would have a biased thought, not to stretch their mind again but to make money from what ever game it is.
I don't think so.
Playing mind games without gambling is more about improve your skill because you're not under pressure.
But, if you play mind games and gambling, it make you play with a pressure because if you lose, you will lose your money, this will affect your mental.