I am aware of the law. I just doubt it applies to elected office, which isn’t a “job” in the full sense of the term.
Do you seriously think it’s a good idea to give Congress the power to add more qualifications to those in the Constitution? Perhaps they could pass a law saying only men are eligible, or only women. Or only people under 5’ tall.
Why not? If they can require SS registration to be president, they can require anything else not specifically prohibited by law, such as a religious test or racial discrimination.
Whether a good idea or not, the site lists very few exceptions and “elected official” is not one of them.
Penalties for not registering can be up to $250K and 5 years in jail.