By any account you read, this ended up being a difficult record for the Stones to put together. They got caught in the trap of trying to keep up with the Joneses, even though when you actually listen to the record, there is no way any of this could be misconstrued as a carbon copy of Sgt. Pepper - not even close - and anyway, a lot of the songs here feel like parodies of the hippie movement.
More to the point, there is a certain feel about Satanic Majesties Request which I find more realistic compared to the Beatles' presentation of psychedelia. Ultimately, on the average, it turned out not to be THE spiritual awakening, and was more like a wacky side-adventure with semi-dangerous side effects - and I think that is the gist of what the Stones get across here, intentional or otherwise.
The whole deal - and 1967, for that matter - put the band in a massive funk, and the only way out was to go back to the roots, as it were.
That said, going back to the album, despite all of these drawbacks and forces working against the group, I do find it to be immensely entertaining as a cohesive whole. Request, much like other psychedelic concept records, is meant to be devoured from start to finish, to really get a feel of what it is about.