Considering his marxist views and that of his family and friends, I’m sure that he would have used that excuse to avoid signing up for SS for a country that he appears to not have much respect for.
From reading parts of his book, not signing up would make perfect sense with his views at the time.
I would tend to agree, with or without a birth certificate status being a factor.
That would make his Birth Certificate and draft form controversies to be two mutually exclusive stains on his history, as opposed to them being linked to a single issue. (assuming both are in fact proven to be scandals)