... It does seem obvious, though, that after Sgt. Pepper they never really operated properly as a band - is that where the myth comes from? ...
I don't think that's a specific / direct cause, but I'd say it was a key factor in setting the context within which the Paul Is Dead meme / UL was spawned. Here's how I saw it and still see it ...
In the space of approximately 1 year (1967 / 1968) the Beatles 'institutionalized' themselves in ways that proved problematic for the four members to handle. This institutionalization occurred in both the musical and business (making a living) aspects of their individual and joint lives.
The release and success of
Sergeant Pepper's put the Beatles in a dominant position within popular music generally. They weren't just figures in the pop / rock sphere anymore - they'd become icons in the history of music overall. The album was so unexpectedly complex and unique it set a benchmark that would be difficult to repeat, much less surpass, thereafter. Once they'd done
Sergeant Pepper's they would have a hard time reverting to anything less epic (as the new
Get Back documentary demonstrates).
Meanwhile, their behind-the-scenes business affairs had reached a scale where they had little choice but to go corporate (founding the multi-faceted Apple media empire in 1968). Once this happened the Beatles were increasingly sucked into dealing with business matters above and beyond the musical activities that initially drew them together.
Things seemed to darken after that. Fans began to lose enthusiasm for their music, because it seemed to be getting too esoteric / artsy / complicated to embrace. It seemed there was an endless series of crises and arguments over Apple affairs, and this sideshow dominated Beatles news.
They'd become mired in their own runaway success.
Near the end of 1968 they released
The Beatles (aka The White Album), which seemed to reflect fragmentation and lack of coherence for the band. It had become clear each of the four Beatles was increasingly veering off to concentrate on his own personal interests and projects. The single they released in the mean time (
Revolution / Hey Jude) didn't make any sense.
Revolution seemed like a snide put-down of the political activism that had emerged during the preceding couple of years, and
Hey Jude was a turgid sappy mess.
Their primary audience (young folks) began to ask if the Beatles were 'over' (i.e., they'd run their course).
This was the depressive context as of autumn 1969, when American DJs and writers dusted off the old (1967) rumor about Paul being killed in a UK auto accident, suggested clues to the rumor's truthfulness derived from the most recent albums, and launched what we'd now call a viral meme.