I think that there is a much better way to have Sports Day. Just fun, non-competitive games and a family picnic after.
At my dc's school, we divided children into groups of 7-10. 2 parents/group.
We had many, many parent volunteers ...and some FUN/ORIGINAL games as well as some expected ones. No races in front of the school. Just fun.
Yes, some dc were awful, but with silly games, eventually everyone was a winner or loser
The emphasis was on moving and enjoying physical activity.
I was friends with the PE teacher and he felt strongly that no child feel humiliated and EVERY child have fun/enjoy physical activity.
I volunteered every year and I still think fondly of those days.
One particularly fun game we played was "kick your shoe up the hill."
I got really good at showing them how to do it. 
He was responsible for teaching generations of children. He watched some pushy parents yell at their dc and shook his head.
He told me, "i remember when those fathers were my students. They were terrible. Now they expect their dc to be better than they were."
I was a horrible sports person bc I grew up on a farm. Nobody to teach me any games. I still feel embarrassed at my being chosen last and being laughed at by teachers and peers.
"Tough love' is a stupid saying. It didn't build my character, it made me avoid sports until I was an adult and finally understood it was lack of experience, not clumsiness that was my reason for poor performance.
When I volunteered at dc's schools, I knew many mothers who told me, "life isn't fair." When they didn't want to provide food/drink for EVERY child at a class party bc we had an odd number of students and we would need only 1 or 2 from a package of treats (hence need to buy 3 pkgs instead of 2).
I told them all this: "it won't be YOUR child learning that lesson will it???" Then, I often purchased the extra pkg. Bc no. It wouldn't be their dc.
I sincerely doubt anyone suggesting that Sports Day is a wonderful Lesson Day at losing was the butt of jokes or worse, their dc was.
Humiliation is NOT helpful. I tried to teach dc how to play most sports, how to lose graciously, how to laugh at themselves, and to only cry once they made it to the car.
But the best lessons about sports was taught by their dear PE teacher who didn't emphasize winning; but rather he taught the importance of enjoying themselves and being kind to classmates.
OP