The original meaning of kid( noun) is a young goat or kid (verb) of a female goat (a doe) giving birth.
It was more recently that kid started being used for children (initially informally but now gaining ground as acceptably formal)
Even now, to kid means you are being naughty or playing some sort of pranks (as in “I was just kidding.”)
Americans started using it because kids (children of goats) are stubborn and they think they saw such stubbornness in their children too.
Though it is now very acceptable and not frowned upon in any way and it is also grammatically correct to refer to human children as kids, I still think it is a way of name-calling your child.
More so, some people like me really have little goats and have children, I think there is a need to differentiate children that are expected to grow into responsible men and women in the future from kids that are expected to grow into goats in the future.