This did not prohibit using the Arizona State form from being used, it just prohibited the state from mandating that organizations use the state provided form and allowed them to use the Federal form instead.
Congress needs to pass a law requiring proof of citizenship on all voter registration forms used throughout the United States, regardless if it is a state or Federal form. Problem solved.
Yeah, that would just make too much sense in a world where citizens matter just because of the importance of citizenship to a stable and prosperous country centered around individual rights. Unfortunately, we don’ t live in such a world.