In what I have read, the fact that John and Robert were brothers seems to have been taken as gospel by the authors, possibly because as Robert's descendants,. they would prefer the link to John Carver to be real rather than unproven.
I like to think that I belong to a parallel branch of the family that had the sense to stay in England. I guess I'll never prove it, but it certainly winds up the American Carvers, who think they are some sort of royalty