I went through jump school at Bragg in 1960. We didn't sing that "funky, funky leg song."
The term 'leg' is a put-down of non-airborne, for sure. The term comes from "straight leg" which describes the unbloused trousers worn by non-airborne troops. Airborne troops created the "pants in boot tops" which than made them non-straight.
That's the history of 'straight leg' as I remember it. It doesn't explain why or how the airborne began doing that in the first place. Oddly, only MPs and airborne were permitted to do the 'pants leg' tuck thing into their boots.
When I was in airborne troops took great pride in how their cap was worn. As you know, we ripped out the 'stiffening' material in the cap so we could push down the ridge. Gave us that cocky look. Which is kinda weird because we called the modified cap 'a cunt cap'. [No offense meant, politically correct ladies reading this.]
As I understand it, all Army wear a beret now. When I was in only Special Forces were permitted a beret. I stoop in a review at Pope AFB when President Kennedy visited and decreed that SF troops could wear the beret 'legally'