This is an extended 12" remix of a couple of previous answers of mine that I've posted here before:

If you look at the British singles charts over Christmas in 1961, you're mostly looking at looking at solo artists. In fact, the only act in the Top 10 which mentions a band is Kenny Ball And His Jazzmen. If you look further down the charts, there's a band called The Shadows, but they're not quite the 'modern concept of the band' - they're an instrumental guitar group (who occasionally backed the pop singer Cliff Richard). There's a folk group called The Springfields (featuring a young Dusty Springfield), and a doo-wop vocal group called the G-Clefs, who wouldn't have played their own instruments. The equivalent American singles charts over Christmas in 1961 is much the same. #1 and #2 are taken by vocal groups, but the rest of the top 10 is solo acts apart from Joey Dee and the Starliters, who were a singer and backing band group.

So, basically, in 1961, where there were bands, they were either vocal groups, or instrumental groups - the modern idea of the band that does both, where the lead singer doesn't get their name in lights (or where there might be more than one lead singer) - was practically non-existent.

And, in fact, if you think about the big rock'n'roll acts of the 1950s, they were all solo acts - Elvis Presley, Little Richard, Jerry Lee Lewis - or singer + band combos - Bill Haley & The Comets, or Buddy Holly & The Crickets. Several of these 1950s acts had a four piece line up like the one you refer to: Elvis Presley's band after 1954 was Elvis (singer/some rhythm guitar), Scotty Moore (lead guitar), Bill Black (bass) and DJ Fontana (drums). Similarly, Buddy Holly and the Crickets initially had Buddy Holly (vocals/lead guitar), Niki Sullivan (rhythm guitar), Joe Mauldin (bass), and Jerry Allison (drums).

However, while if you buy a Buddy Holly greatest hits CD today, it's probably either credited to 'Buddy Holly' or 'Buddy Holly and the Crickets'; this was not how their music was released in the late 1950s. Instead, because Holly's producer Norman Petty wanted to maximise the group's sales potential, he wanted to release singles on more than one label because an individual label wouldn't promote more than one song by the one artist at once. Because Buddy Holly & The Crickets' music was being released on two separate record labels, their releases ended up being released under the name 'Buddy Holly' on Coral Records, and 'The Crickets' on Brunswick Records (the arrangement was that if the record had vocals from just Buddy Holly, it was credited to Holly; in contrast, if there were group backing vocals, it was credited to The Crickets).

So the idea of The Crickets (as opposed to Buddy Holly or Buddy Holly and the Crickets) was something of a contractual obligation, but nonetheless, people in Liverpool in England wouldn't have known that; they just would have seen a group called The Crickets that both sang and played their own instruments, and were inspired by example.

And when I say people in Liverpool, I mean The Beatles. They have talked in interviews about how much they loved the idea of The Crickets (this is discussed reasonably extensively in the quite scholarly Beatles history Tune In by Mark Lewisohn). According to Lewisohn, the group had arrived at the idea of calling themselves The Beatles in May 1960. Lewisohn quotes George Harrison as saying:

"all the groups were ‘Harry and the Somethings’ or ‘Charlie and the Somethings’ so we decided we’d just have ‘The ——’ and that’s it. Instead of trying to get one of the fellers as the star with the rest in the back we thought we’d just all be together."

When Pete Best joined the band in August 1960 on drums - he'd basically been hired so the band could play shows on an upcoming Hamburg tour - one fully-fledged version of the modern idea of the 'band' had effectively come into existence; it would take another couple of years for the group to release their first single, 'Love Me Do', in October 1962.

Hawthorne, California also saw another group that had separately arrived at the modern idea of the band about the same time as The Beatles: a group called the Pendletones, which had a four-piece band line-up - Al Jardine and Carl Wilson on guitar, Brian Wilson on bass, and Dennis Wilson on drums - plus a lead singer called Mike Love. Like the Beatles, all of the group sang as well as played their instruments. The Pendletones recorded a track called 'Surfin'' in October 1961 and discovered when it was released on a local indie label Candix, that their name had been changed to The Beach Boys.

The Beach Boys didn't quite come to the modern band idea the same way as The Beatles; instead, they were simultaneously a vocal group and a surf guitar band, and both doo-wop vocal groups and surf instrumental groups tended to be called The Somethings. Their first album on Capitol in 1962, Surfin' Safari, features both guitar group instrumentals like 'Moon Dawg' and doo-wop-esque vocal group songs like 'Little Miss America', and several songs that do both at the same time.

The Beach Boys and The Beatles had very successful years in America and Britain, respectively, in 1963. Record companies noticed and tried their best to emulate that success. The Christmas 1963 British singles chart features bands called The Rolling Stones, The Swinging Blue Jeans, and The Hollies and Fourmost. The Christmas 1963 Amiercan singles chart features not only The Beach Boys but also The Kingsmen, and The Trashmen. Within months, Beatlemania would hit in the U.S., and bands became very cool indeed.

By Christmas of 1964, the U.S. chart would be infested with British bands modelled after the Beatles: The Zombies, The Rolling Stones, The Kinks, Herman's Hermits, Manfred Mann, The Searchers, and The Animals. And in 1965, these British groups would be joined at the top of the charts by American groups like The Byrds, The Beau Brummels, The Knickerbockers, The Sonics, and the Lovin' Spoonful.