So, having read this through, it seems that:
Nobody is denying that there are VVT jazz basses constructed in 1961.
But these would have happened later in the year and that anything manufactured in, say, August 1961 should treated with caution.
(I'm only getting involved as it's me who posted the link to the guitar that caused all this hassle. Personally, I think enough doubt has been cast on its originality that I would seek a second opinion from someone who can see the instrument in person, but I'd be prepared to do that if I were going to drop £18k on it, given my experience with vintage dealers.)