If you want the Beatles without George Harrison, you probably need to have Paul and George have a falling out in the fifties. Paul wanted him in the nascent group he was in with John, and you absolutely cannot have Lennon and McCartney have a falling out, as they very well could.
This opens a hell of a lot of butterflies. But had McCartney never extended an invitation for George to join the Quarrymen, I do not think there was anyone waiting in the wings to play guitar, especially as both Lennon and McCartney played the instrument. During their earliest period they needed a bassist and a drummer, not an extra guitarist. George was a convienent addition, but they would not go out of their way to find someone else. The group's final form may well be something of a glorified duo, if they make it at all, what with butterflies and all.