It is true that the Rolling Stones generally copied The Beatles in just about everything, or more accurately did a take on them. It is also true that it took about 4 or 5 years before they found their own style and feet. They were a copy cat band who made it big and had longevity.
Here is a clip on Youtube with John Lennon criticising them, after Jagger made some disparaging remarks about The Beatles after they split in 1970. Lennon never pulled punches and effed and blinded along the way. He said they would have got nowhere without The Beatles and that they even gave the Stones their fist No. 1 hit. Lennon and McCartney were in the studios when they recorded it and helped along and even rewrote the song on the fly moving to the side of the studio changing the words to those that they thought would fit the Stones' style.
It's not true at all that the Stones "generally copied The Beatles in just about everything, or more accurately did a take on them". I'm sure the Stones were influenced by some things the Beatles did to a certain degree, but who wasn't influenced by the Beatles in the Sixties? They were a huge phenomenon. Despite being influenced by a lot of the same artists as the Beatles like Elvis, Chuck Berry, Buddy Holly, etc, etc, the Stones had very deep roots in American Blues music with such artists as: Howlin' Wolf, Muddy Waters, Little Walter Jacob, Jimmy Reed and Slim Harpo. Blues music was the staple of the Stones early sound. You did not hear much of a Blues sound in Beatles music until the late Sixties with albums like The White album, Abbey Road and Let It Be. One of the reasons why the Stones made it big is because they had a different sound then what was popular at the time, which was the Mersey Sound: Beatles, Gerry & The Pacemakers, Billy J. Kramer & The Dakotas, etc.
I don't know exactly what Mick Jagger said the ticked off John Lennon, but I think he went a little to the extreme in this interview with these comparisons. I wouldn't take what he says as gospel fact by any means. And the Beatles did not give the Stones their first # 1 hit. You are referring to "I Wanna Be Your Man" which was the Stones first Top 20 hit in Great Britain. One thing I will say is the Stones version of "I Wanna Be Your Man" sounded nothing like the Beatles version whatsoever - so it can hardly be called a "copy".
John Lennon said most. I have no need to duplicate. In the early 1960s sweet nothing was coming out of London. The music scene there was dire. Many in London thought the same so to be "hip" went for off the mainstream to blues. The Stones did many US blues songs with not much of a different angle. It was just white boys singing US black music. It was only when the Stones were raucous that people took note. The Stones were a one dimensional band focused on one lead singer. The lead singer took most of the credit. They were very limited however in their narrow compass did well.
BTW, the Merseybeat sound evolved. By 1965/66 many of those bands sounded very different maturing greatly. Listen to the Merseybeats, later Merseys. Also, I Wanna Be Your Man was written by the Beatles. Lennon and McCartney were in the studio when they recorded it. John Lennon was always disappointed when band did cover of their songs that sounded just like the Beatles. he liked a different take on their songs. The Beatles would have greatly discouraged the Stone to do a Merseybeat type of treatment.