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.
I stand by what I meant to say and you say it better than I ever could even if we disagree on one single point - the comparison between Pepper and Majesties. But that's a discussion for another day...