> solving game puzzles doesn't necessarily mean you get better at school math
That's not how improving through games works at all.
The things you understand through games are much more general than the surface of the gameplay.
Total War (or Starcraft) won't teach you anything about commanding armies. But if you're observant enough it will teach you a difference between level-1 versus level-2 strategies. It will definitely teach you about the impact of timing on execution. It will show you complexity, risk, loss aversion, and how battles are usually won or lost before the first shot is fired. All these are useful things to experience so that you are better equipped to deal with them in real life.
What's even more important, the games will show you how YOU relate to these intangible concepts. How loss averse are you? Do you naturally tend to maximize win or minimize loss? How easy it is for you to abandon a pre-established plan? These and more are insights into your own nature that are not easy to get.
Finally, the multiplayer games will show you the human nature. You'll understand, for example, that different people play for different reasons, and just this understanding alone was worth all hundreds of hours I put into M:TG. I guess that most of the multiplayer insights are also available through participation (and/or managing) a regular sports team but I would have never been able to join as many of those as I had gaming teams.
TLDR: Games guide the player to instinctive understanding of categorical truths that underpin the simulations, and that is only possible through countless repetitions of similar scenarios* in different contexts. This is the true value of gaming.
[*] I'm not talking about grinding here[**], but playing same or similar games many times.
[**] although grinding can teach one about how seemingly small process improvements somethimes add up to a qualitative break ... and sometimes not. Figure out the best way to do a cow level run to learn more!