That's a fantasy constructed by people who want to keep idolizing their favorite musicians. Labels do exactly what the musicians signed with them want them to do. Heck the vast majority of record labels are owned by musicians.
Indie labels and the occasional hit independent label like Def Row/TDE are small in comparison and usually only include the owning musicians established music, not their beginnings. Bands often have to fight to buy back rights for that kinda stuff.
Interscope, Reprise, etc are all owned by the big three. Interscope wasn't even founded by artists.
Also labels (like Def Row and Reprise) often sell out to the big three, so even if you sign to one of them when they're independent, that won't mean it'll last forever and you as an artist won't get veto over that.