C.E. Evans wrote:
I think it wasn't until after the Enterprise-D's encounter with them that the Borg became known to everyone and ceased being a rumor known to just a few.
I agree with this. It's entirely possible there were rumors swirling about cybernetic beings from another part of the galaxy, but without any proof of their existence (as far as Picard or anyone else was aware), there would have been no reason to take the rumors seriously.
Rumors about strange aliens crash-landing on Earth would be a big deal in our society because we've never encountered anything like that (that we know of), but for Starfleet officers whose lives revolve around all sorts of weird things, rumors about a race like the Borg wouldn't be much of anything to get excited about. Things changed after "Q Who?" because Picard realized the significance of the Borg threat, but to suggest that he'd have known anything about the Collective before is hard to believe.