Perhaps. Perhaps not.
The BC is the foundation of everything. He has to produce it for his school records (his is the only BC missing from his kindergarten class), his driver's license, passport(s), etc. Every piece of paper he put his dob on would be subject to inspection. He can't have that.
OTOH, if the BC was the only document that was sealed, then even CNN would be zeroing in on it. As it is, with EVERYTHING under lock and key, then he can play the BC off as a red herring and who knows what the real zinger might be. He's laughing at us because he knows there's no way anyone can get through 50 years of documents as opposed to just the one bc.
All that aside, daddy was Kenyan/British making the usurper ineligible.