Yes, they did indeed popularise psychedelia, or a generic version of it, but they were hardly Pink Floyd or The Pretty Things, let alone any of the west-coasters. They capitalised on an existing "thing", and made a commercially-successful fist of it, but it was hardly "cutting edge", as psychedelic music went.
The Beatles were a pop group. It was cutting edge pop. And the pop genre in question was one they helped create.
The West Coast scene was based on live performance - much of it was a deliberate reaction against studio pop. Some of the music itself was a little conservative and initially it didn't record well. Nonetheless there were groups like The Byrds who to begin with were enormously influenced by the Beatles and in turn went on to directly influence them back.
The original 'underground' scene the Floyd played for in London was tiny by comparison to the West Coast scene - closer to a club scene. The distance between it and the rest of London pop culture wasn't that large. The Floyd began recording pop singles (sometimes in the studio next door to the Beatles) before they decided they didn't want to be pop stars and did something else. But commercially that possibility to do something else only existed because people like the Beatles had shown there was money to be made in allowing groups a degree of control over what they recorded.
We'll never know what psychedelia would have been without The Beatles, but we can extrapolate that given the strong counter-cultural currents of the time, it would have been different, perhaps more "scene", less commercial. What we do know is that even before The Beatles burrowed into psychedelia, British and US bands were producing fantastic and popular music.
I sometimes think people view the psychedelic underground in 66 and 67 as if it was as large and as clearly differentiated musically as the 80s/90s rave scene. It wasn't. Initially it was very London-centric, and as a crossroads at which various already existing musical, bohemian and avant-garde scenes met and cross fertilised, it was musically quite diverse. A much broader counter culture (or perhaps more accurately set of countercultures), within which distinctions emerged between 'pop' and 'rock' and stuff which wasn't either, then developed and very rapidly spread nationwide. Initially one of the transmission belts for that spread was pop psychedelia. I'm not even sure that a distinct UK 'psychedelic' music would have developed without the pop element - not just the Beatles of course but the mutations of mod and freakbeat and everything else. At the start it was the glue which held it together.
So I don't accept that the Beatles were parasitic on psychedelia. The relationship was symbiotic. Although they didn't play for the 'underground' scene in London Beatles money supported a number of the elements of it. More importantly, beyond any contribution they brought to the table themselves, the Beatles and all the other pop psychedelic groups were a gateway drug for an enormous number of people who would have been entirely untouched by what was happening at the UFO club or Middle Earth or the Arts Lab.
Speaking for myself, if I hadn't spent my 1967 Christmas money on a Woolworths transistor radio and had it clamped to my ear for as much of 1968, my perfect pop year, as I could afford batteries, I wouldn't have also discovered the stuff Peel and others were playing. (One legacy being that while as a teen music nerd I later subscribed to all the usual sorts of snobberies and inverted snobberies I managed not to lose the ability to enjoy pop. I love art but sometimes only fuck art let's dance will do. Don't see a need to prioritise pop or culture, art or entertainment. I want it all).
Love Rob Chapman's book about Psychedelia - thought of just posting the chapter about the Beatles
but I can't be arsed.