Do I get it right? Your DH's crime was not wanting to be involved in "your argument" but when pressed, he acted as a father and took DD's side, while you had acted as the organiser and missed the opportunity to be a mother when your DD needed one?
My DC compete one at a state, the other at national level in two different sports, and I see a lot of mothers like you, sorry. Super involved, really thriving in the attention and gratitude of other parents, loving they are being told their job is so important and how nothing could happen without them, and so on, but then during the competition they are so critical of their child, their poor attitude, sportsmanship, when they don't realise they have inflated the whole process.
Your immense work ended up benefiting another girl but not your daughter and I think deep down you are hurt by this and resentful of your daughter even if you are not aware of it, and this is why you were short fused with her.
And your DD was right to cut you short, because she did know what went wrong and that it was her fault, so interrupting you in your wrong interpretation was the right thing to do. Calling you an idiot was wrong from her, but she was trying to tell you something and you didn't listen and told her to be quiet.
It was a bad day for everyone and you dragged your DH in the mud when he wanted to stay out of it.
Your involvement is this competition was greater than the competition itself. It fulfilled a sense of purpose, gave you joy, and was an escape from your brother's issues. The pressure on your DD was probably sky high, because she wasn't only competing as herself but as the organiser's daughter as well.
I actually feel sorry for her. It was her day not yours.
I hope tomorrow will be a better day and that you can move on.