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).

Weightlifting trains balance extremely well. A max effort power snatch for example requires excellent balance.

Are you kidding? Have you ever snatched heavy? thrown around kettlebells ?

I'm not an expert, but handstands and similar skills might require more precision and whole-body balance than any weight-lifting exercise?

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.

Compound exercises do this. Flexibility can be improved with stretching.