What? A good weight lifting routine which incorporates squats, deadlifts, dumbbell rows, bench presses and little more, will absolutely strengthen every single part of your body. As for mind benefits, I can't speak to the scientific evidence of either, but in a physiological level I don't see why weights should be any worse.
There's a difference between not touching something and training is less. Weightlifting touches on all three of those things ("flexibility" is usually called range of motion w/r/t weightlifting though).
Balance is heavily trained with heavy weights as well because now, not only do you have to move and balance yourself, you need to do it with a heavy weight on your back for example.
It's hard to compare the two though because one might day that, in old age, being able to do a handstand is less important than being able to haul groceries along uneven terrain standing up.
Yes, people who do yoga regularly have better balance in various odd positions then weight lifters. Yes, they do learn balance tricks (like walking on slack line) faster then weight lifters.
