In general, better physical workout and diet does genuinely improve brain function.
Hard for the brain to function well in a 500lbs diabetic body...
Brain training does work for some people; around 20% of the population won't gain anything from it, regardless of what you make em do. And even for those that works it has limited scope on the benefits.
Everyone really should try it. I firmly believe there is nothing that you can do in 20 mins a day that will improve your life more.
If I wanted to (at least try) improving cognition, I would spend hours reading math textbooks and doing exercises, learning languages, perhaps reading about philosophy in my off-time; not strafing left and right with my WASD keys and trashing on kids all day in Counter-Strike.