“Registering for Selective Service is in no way an eligibility requirement for the office of President.”
Where did you get that information? It is totally wrong. Failure to register with the SSS means you are ineligible for ANY Federal job and many State jobs.
For this to apply to election to president, it would mean Congress can add additional requirements to those in the Constitution. I doubt this would be Constitutional.
In fact, I doubt it applies to election to any federal office. Such are not "employees" in the strict sense of the term.