In the case of The Beatles and Elton John, once they became enormously successful, they started their own record labels. The Beatles created Apple Records.
And what a nightmare that turned out to be.
The Beatles were still under contract to EMI at the time Apple was founded, so EMI continued to own the recordings that came out with Apple labels on them and agreed to distribute Apple product until that deal ended in 1976. By that time, the Beatles had been suing each other for six years trying to shut Apple down and end the bleeding.
Elton John formed The Rocket Record Company. Elton's record company signed Neil Sedaka and, if I remember correctly, Heart.
Sedaka yes, Heart, no. They went from Mushroom to Epic.
But I don't think Elton ever recorded his own albums on his own record label, for reasons that aren't clear to me.
BLUE MOVES was on Rocket, as was his duet single with Kiki Dee, "Don't Go Breaking My Heart". He went back to MCA after BLUE MOVES until 1996, when Rocket, which had been dormant, was re-activated. Four of his albums (MADE IN ENGLAND, THE BIG PICTURE, SONGS FROM THE WEST COAST and PEACHTREE ROAD) were all on Rocket. Elton moved to Mercury after that.
This makes me wonder if Richard or Karen ever considered starting their own record company. From what I can tell, it never entered their mind.
They had enough trouble without owning their own label.
I'm having trouble thinking of any that were more than an imprint (Rolling Stones Records, for example) that had long-term consistent success (A&M doesn't count---but Almo Sounds would).
Reprise was a disaster when Frank Sinatra owned it---which is why he sold it to Warners. Maybe Madonna's label, Maverick, which had million-sellers from not just Madonna, but Alanis Morrisette. And Swan Song, founded by Led Zeppelin, which had them, Bad Company, Dave Edmunds, Sad Cafe' and the solo output from Jimmy Page and Robert Plant.