Instead of citizenship at birth (for anyone) do permanent residence for 18 years and then they can apply to have that extended for 10 years, or they can apply for citizenship. It is already the case that your US citizen parents can apply to get you a passport, if you are under 18 and not a citizen.
Make citizenship a formal, active process, based upon points, for everyone. If both of your parents are a citizen, you were born there and had your primary residence there your whole childhood, you get 100 points. And then scale down from there. Then raise the cost of citizenship by requiring every citizen to pay a high poll tax or sign up for the draft or be in public service. Or something like that.
And then see who signs up.