Organization = legitimacy.
I do not know of any cons.
and yes we belong to one. It traces roots all the way back to funakoshi. My senseis were students of Mr Nishiyama who were student of Mr Funakoshi.
If you weren't part of an organization and someone questioned who you were and what you do, you would say exactly what you just did; "my senseis were students of Mr. Nishiyama who was a student of Mr. Funakoshi." You answer lies in your lineage, not that you art part of some larger association, again it is that student teacher relationship.