Nuts, I have no idea what you're talking about. Your children are not quasi citizens. Their birth certificates with you, a British citizen, listed as the father, are sufficient proof that they are British citizens. If they are travelling outside the UK, they would need to travel with British passports into the UK just as any British citizen would, but if they were to live in the UK they wouldn't need a passport any more than anybody else. Even their expired passports would be proof of their citizenship. The only difference between them and British citizens born in the UK is that their children will need to be born in the UK to be British citizens, as citizens by descent cannot pass on their citizenship to the next generation unless in the UK. If Juliloew's father had been British instead of her mother, she would be a British citizen as fully as anyone else who is British, and just be able to apply for a passport without having to go through an extra step of paying a lot of money and proving she is 'worthy' of citizenship. It's total nonsense!
As for the US case, if you are born of a US parent, you have no choice but to be a citizen of the US, and must use a US passport to travel to the US.