yeah, i would say that it does improve body awareness, or at least it has for me. learning completely new skills from a baseline of zero means learning totally new body organization, and that means you're clumsy and have to pay close attention to what you're doing. i've always had decent kinesthetic awareness: i'm an okay athlete, can pick up the basics of new sports pretty easily. but for example i knew my handstands were sagging even before i saw those videos the other day, and that's because i was paying close attention to my whole body.
once the basic skills are ingrained -- pirouettes, kip-ups, handstands -- i have no doubt that will translate to improved balance. one thing i like about the GMB program (they have something like 10, actually, and i'm doing Floor 1) is that it's cumulative. you start easy and add complication in a clear way.
as far as moving better in general, too early to say. one thing i will say is that my wrist mobility has definitely improved, and i think shoulder mobility a bit as well. wrists don't play much role in overall movement but i'm a believer now in their commitment to stretching and mobility, which i'd kind of drifted away from for the last few years.
