I don’t agree in general that’s how the system works or how most Americans think. But again that has nothing to do with the OP. I thought you were asking about American culture and I wanted to clarify.
I think once they’re citizens, and id say generally foreign born retain the $origin-american but their kids don’t always. I don’t think you typing in “purely a matter of skin color” is writing angry posts into the void that have nothing to do with the OP.