There are exceptions, of course, as to whether or not the children acquire US citizenship at birth. The mother's absence from the US will pertain here. I don't think there's any doubt that the children may opt for US Citizenship at age 18, if they can make it to the States.
The main issues here are that 1) - Saudi doesn't recognize dual citizenship to begin with, and 2) - they don't recognize any rights except Saudi male rights. It'll be almost impossible to get the kids out of the country - they can't just walk up to a airline counter and hop on a plane there.
And the exception is that one is not born in the US.
I believe you owe your readers a retraction of your statement that "Of course they are US citizens".