I do feel for your DS and I do hope you get some satisfactory answers from the school.
School PE put me off sport. I was "that" child too. I wasn’t overweight, but I certainly wasn’t athletic and I hated team sports. Our teachers believed in choosing the two sportiest girls in the class and leaving them to select their teams from the rest of us. Along with one other girl, I had the ritual humiliation of always being picked last and the person who ended up with me scowling over it.
Thankfully, I don’t recall any laps of shame on the sports field, but everything else about PE was just horrendous. It still amounted to public humiliation because of the utter obsession with competitive team sports, at which I was useless.
It was sadistic too. I remember us all being made to do hockey and netball in the snow without being able to wear tracksuit tops or bottoms. Just shorts and a t-shirt. Even the sportier ones disliked that. Teacher would be standing by the pitch in her tracksuit and a big outdoor coat.
If your DS was punished for not being one of the faster runners then that is humiliation and needs to be tackled.
Is this one of those PE teachers whose favourite catchphrase is somewhere along the lines of "there’s no such word as can’t?" Mine used to come out with that bullshit.
I was able to give up school PE when I started my O Level options at 13 and it was such a relief. We were actually supposed to do some PE for that couple of years but somehow it never made it onto my timetable and I certainly wasn’t about to draw attention to that. I got away with it. School didn’t notice the error until I was about to go on study leave. 