Character building? Snort! Perhaps character can be built through a certain amount of adversity. But I think it's under estimating the fear. Like saying a little mild torture is character building too... water boarding, sensory deprivation, sleep deprivation, that kind of stuff.
In reality the worse type of PE lessons can break children, and decimate their self esteem, body confidence and emotional resilience, not just at that moment but for years after.
Sorry if that sounds extreme, but what (used to?) happen/s in PE IS extreme. Proper lord of the flies stuff but with the addition of an adult sanctioning it.
IME it teaches you that people in authority have the power to bully and humiliate you. And you are powerless to defend yourself or keep yourself from harm.
It teaches you that they have the power to take away your body autonomy and that listening to your own body is not allowed.
Listen to the evil fucker in the nice warm tracksuit blasting away on their whistle and shouting abuse at the fat/ thin/ asthmatic/ poorly / disabled / puking / vulnerable stragglers. Don't you dare listen to your body, not even if it's doing you harm to carry on. Especially not then in my experience! Teacher knows best, and if you cannot physically obey then you deserve what you get, right?
It teaches you that they have the power to make sure you know exactly where you are on the pecking order of life. By encouraging other bullies... I guess you could think of it as a bullying master class.
It teaches you that people can laugh at your body and its ok to make fun of anyone who looks different for whatever reason (boobs, hair, expensive kit, thinness, fatness, red hair etc etc etc).
It teaches you to do anything you can to escape. And that a short sprint and a one handed jump over the fence is the way to freedom, then yay, let's get physical! And if that's smoking fags or hanging out by that dodgy street with the scary men hanging around... so be it. Or hiding in a field in the rain all day, then yup, even that's better than the systemised cruelty of your traditional cross country / PE lesson.
Auden had it right.
Now, if they actually, taught, that might really help children get fit and keep healthy!