Not everything that is wrong is therefore unconstitutional.
The Constitution does not say, for instance, that convicted mass murderers or traitors are ineligible for the presidency, on the possibly inaccurate assumption that the voters would never elect such a person.
As it is, AFAIK, Charles Manson is eligible for the office. Can’t vote, in most states, but I think he could be elected president, as far as the Constitution is concerned, anyway. He might be ineligible to be on the ballot in some states. Although a TX incarcerated felon was on the Dem primary in WV and got a bunch of votes.
Obama would still qualify as a presidential candidate, regardless of the alleged SS# fraud. If anything ever came of this, it would be handled through impeachment.
But crimes and misdemeanors are impeachable. This is a pretty serious felony for the average American. What is your basis for giving the President a pass?