If this is true, there is surely a way to present material honing that specific skill as a game or series of games that would generate more intrinsic interest from more kids than math does.
I don’t think the methods of thinking per se are why we teach math, though. Might be part of it, but if that’s all, I think we could do a lot better for a lot less effort for all concerned. I think it’s because the math itself is useful. If in fact the point were to teach methods of thinking, I doubt we’d teach it as we do math—why would we, when it generates such resistance and loathing from so many students?
For example, the way to solve a quadratic is to reduce it to a form one knows how to solve (via competing the square).
The specifics of the mathematics are not the prize, the methods of thinking are.