Their status is in doubt. Does your nation of birth extend citizenship to your children, or even grant it automatically? Have they accepted this citizenship if offered? If so, then they’re not. They’re eligible for everything else, with the exception of being in the direct line of Presidential succession.
Well it’s an interesting thing you should ask that. Because my country of birth is not my country of citizenship. I am a citizen of my country by descent. As a direct result, my children are not eligible for citizenship to my country, because the right to citizenship by descent only extends one generation and that was me. US citizens is what my children are, and it is all they can be. So tell me again why it is they can’t grow up to be president?