Yep. The Stones have said that early on the Beatles encouraged them to write their own songs.
The Stones were a different animal than the Beatles but the Beatles provided some inspiration for the Stones at times - notably Her Satanic Majesties Request. The Stones were like the Beatles’ evil twin. :-)
Perhaps the nadir for the Stones. The Stones have done a bunch of stuff that's not great, but HSMR is just bad.
It’s kind of funny. The Beatles were from Liverpool, which is a pretty tough port. The Stones were college kids from London.
Branding is everything.