I would focus on getting the kids moving in whatever way they wanted. Even if it's playing tag. Not sports. The kids who want to compete can play outside of school. Not everyone enjoys team sports.
Forced team sports and assessments in front of the other kids are seriously brutal and humiliating sometimes for kids who aren't athletic. Speaking as someone who sucks at team sports, I remember many miserable PE lessons spent getting yelled at for an hour by my teammates for missing a ball or whatever. There were only two units in the whole year that weren't dreadful- swimming and distance running- because people actually wanted me on their teams/I was better at those things than a lot of the class. The rest of the year was spent basically having a panic attack before every gym lesson because I knew it was going to suck. Kids can be pretty awful. Hell, so can some gym teachers, though luckily mine were always great. Honestly, I know some people think it's important to measure fitness, but as someone else mentioned it's often very very demoralizing. I think typical gym classes just teach kids to hate exercise and sports.
In my last years of high school we had a bunch of different options for sports in gym, so we could pick something that we actually enjoyed. There was even stuff like yoga, dance, we could go for a run alone, lift weights, etc. But I'm not sure how that would work in elementary school, with the large class sizes, need for more supervision than 16/17 year olds, and limited teaching staff.
In elementary school I had a teacher who would take us down to the gym each morning before classes to do basic calisthenics, play tag, stretch, and just run around and burn off energy before sitting down all day. Totally non-competitive and she didn't single anyone out for not being very good at something or call out scores in front of the class (she would keep a private record of how many sit-ups etc we could do, but that was just for our individual benefit). She was totally chill about everything, and told us that the main thing was just to get moving. It was the best part of the school day, usually. Something like that would be awesome. Not sure how feasible it would be for a lot of schools though.
Edit: Christ, sorry for writing a novel. I'll do anything to avoid studying for organic chem, evidently.