It's all part of learning how to cope with life - some you win, some you lose. The child in the OP is learning from her mother that, if you're not going to be first, top in everything, don't even try. Life's not like that which she will find out later.
Despite me advocating sports at school, I was hopeless at running and athletics but, oddly enough, good at tennis and netball.
Did I feel humiliated because I was the one who kept dropping her egg off the spoon and never won? No, I don't remember that at all, I was probably giggling with a friend at the back.
Sports days should have something for everyone and good teachers should make it fun.