Of course. But not that many are successful at it. Even the Beatles floundered for years before success came only after signing with a label. Note that Ringo was a session musician brought in by the label. George Martin made enormous contributions to their music.
Their original drummer, Pete Best, could not keep time when recording. George Martin raised the issue that he needed to be replaced, that's when they decided to reach out to Ringo.
- Those bowl cuts came years before any success, specifically during the summer of 1960 in Hamburg from a friend of the band.
- The suits were insisted upon by their manager, not the label, in an effort to help get them signed.
- When they first got attention they were doing and playing exactly what they wanted to, mostly because they were singularly focused on being successful. Their set list was mostly comprised of ~500 popular cover songs from the time. You can hear much of this on several of their early albums which were heavily loaded with these - those were the same songs you could find them doing back before they made it, that they played in the Cavern Club and strip clubs in Hamburg. George Martin was the one who pushed them to be bold with the material, get creative with the arrangements, etc, and then the London scene pushed them further forward from there.
Their pre-label original tunes just weren't that good. They played mostly covers. Can you suggest a pre-label Beatles original tune that was worthy?
They never brought in session musicians to play their own pieces. All guitar, drum, bass, piano, etc parts were played by the members of the band and on very few occasions piano pieces by George Martin.
Eric Clapton was not a "session musician", he was doing an uncredited solo for his best friend George Harrison's song.
George Martin did a lot more than just play a few piano bits - he had a lot of creative input. Here's what "The Beatles" by Spitz has to say about it:
"Two weeks later, on his way to the Studio from Sussex, scheduled to give the song another shot, he was explaining to Eric Clapton how something radical was needed to light fire under the Beatles. "We were in George's car, driving in London," Clapton remembered, "and he said, 'Do you want to come and play on this record?'" It was an astonishing invitation. The Beatles had used plenty of Session musicians on other albums, but no one capable of upstaging them, certainly never rock 'n roll virtuoso on the level of Eric Clapton. Clapton hesitated, unsure of what to do. He knew the other Beatles "wouldn't like it," but George brushed aside his reservations. "It's nothing to do with them," he insisted. "It's my song, and I'd like you to play on it." Before anyone had chance to object, Clapton was already in Studio Two, strapping on his Les Paul guitar and listening to the rhythm track mixed down from their work on the sixteenth. The song was pretty much there, creating an effortless, affecting groove, but it lacked dramatic device to liberate the emotional tension that is never far from George's caged expression. Clapton's poignant guitar riff provided everything it needed. The way it weeps and moans, held in check by Eric's incisive phrasing, creates the longing that gives the song its emotional center. George's vocal couldn't have been more enchanting as he squeezes the mournful lyric of all its desperation, until by the end, he seems to be just barely hanging on, just riding atop the surging guitar as it works to strangle his overlapping cries."
The "just a performer" model is only true for that small percentage of singers who are just performers. That model includes some of the bigger household names. Some of them are essentially fashion models who sing and dance, not musicians. Others are singers who also produce, and have a feel for what sounds good even if they can't play all the parts. (This is harder than it sounds.)
There's literally an entire industry of independent artists who record and produce at home, do their own tour management and logistics, maybe mix in a studio for some final polish, handle distribution, and so on.
And an entire related industry of bloggers, DJs, and other online sites that review and curate the most interesting new music.
The labels keep on keeping on, but their business model is very fragile now. They wait for a producer to manufacture a band, or for a band/artist to bubble up from the indie (artist - not musical style) scene.
When they reach a critical mass they try to buy them. Sometimes this works, sometimes it doesn't.
The bizarre thing about A&R budgets over the last few years is that they're as huge as they've ever been, but concentrated on a relatively small number of artists. Because there's a lot of money involved the labels are ridiculously risk averse. If a signing doesn't turn into a cash cow after the first album they're dropped.
This has always happened, but the cycle is so short now that only acts that have instant mass appeal or a guaranteed existing fan base are successful on labels. Everyone else is indie.