Oh definitely. John's love and respect for Chuck Berry, and Paul's love for Little Richard is apparent, even in some of their later works.
The Stones kept it simple. I actually hated the stones growing up. Because they weren't very intricate. Now I love them for the same reason. Just simple fun rock, blues and country. They're basically an American band that happens to be from England.
And for what it's worth, Keith's favorite non Stones album is Powerage.