My experience has been that mental math training, while not directly useful, generally seems to improve the way my brain works and enhances my analytic thinking

My experience has been that mental math training, while not directly useful, generally seems to improve the way my brain works and enhances my analytic thinking. This effect is a lot more tangible with longer, multi-step problems. In college (physics major) I made a point of doing every integral in my head if I could, and over time I found that my mental 'scratch space' seemed to be getting longer - that I could hold more in my short-term memory at once. I've also observed that being a person who is 'good at math' approximately correlates with being a person who does math in your head. [Related observation: that students strongly segregate into "good at math" vs not, as though many people are completely incapable of a mode of thought that comes naturally to others. I think it's trainable, and people who never do any math, especially in their heads, are specifically not training it.] These are entirely anecdotal and I'd love to find more concrete data, but, right now I would absolutely vouch for mental math training.