I had always heard exactly that. They brought him in because they knew him from touring in Germany and they all liked him and knew he was good. Even in the ads/previews for this documentary, it talks about how they “brought him in” to play with them, not that he wandered by to say hi. I guess there was no reason to think otherwise, because the truth seems a bit hard to believe.
I still find variations out there. This one seems plausible:
During his break from the group, Harrison caught a performance by Ray Charles in London. On the stage that night was Preston, whom Harrison and his fellow Beatles had befriended during their years in Hamburg. Excited to see his old pal, Harrison sent Preston a message inviting him to join the Beatles in the studio.
So he “stopped by to say hi” after George invited him to do so.
