Their arms are not in 'better' shape, just different. Possibly even in better shape.Many of the long-time boxers I've seen have massive arms with tiny little stick-legs. It's not so much that their legs aren't in good shape, as it is their arms are in way better shape
Large amount of elite level pro-boxers have smallish, really compact, ridiculously conditioned legs for maximal mobility in the footwork, to be as fleet on feet as possible, with ability to keep it up for long time.
Just adaptation. Most of them with good, classic form just don't need thick legs at all.
'Kick-boxing' is really broad term for pretty much anything that involves all around striking. There will be plenty of elbows and knees, depending on style and school.I thought they might mean Muay Thai as well, but in both instances they also referred to it as kickboxing and in neither of them did they mention use of elbows and knees, which is one of the key differences between Muay Thai and kickboxing, if I'm correct.
AFAIR "American kickboxing" competitions are pretty damn restrictive, generally with striking with fists and feet above the hip, indeed, but that's only one style of competition.