Incidentally, Let It Bleed was an answer to Let It Be, whatever the release dates of the albums. What else, a coincidence? I don't have my reference material handy, by you may recall that Let It Be was first a single and that this was the album, wasn't it, that was recorded earlier (before some other Beatle work) and released later. They were quite aware of each other's work and in light of that album release dates are irrelevant. Also, much of the Stones work up to that time was in response to the Beatles. This was more evident in England where this rivalry was very much up front. Where do you think Their Satanic Majesties concept came from?
That was their biggest goof, to be sure; but rivalry is a misleading term in the Beatles/Stones context. It was more a call-and-answer deal ON THE PART OF THE STONES who, like all bands of the time, used the Beatles as a reference point.
The Stones were so close to the Beatles in digestive resources as to be on ALMOST equal footing (they specialized, the Beatles didn't), and the interaction produced what is a great era in music (to my mind, the best).
Certainly Majesties is laughable compared to Pepper; but Bleed can hold its own with anything. If it was, indeed, a riposte to LIB, so much the better: the Stones synthesized their times in a way that the Beatles (relative lunkheads that they were) could never do.