I think history counts for a lot in this. (Of course, I always think history counts for a lot as the answer to every question, so feel free to grab your grain of salt, which I hope you carry around with you for emergencies.)
Start with the 1950s. The U.S. had a mainstream culture of singers of standards. You could find some groups who harmonized while singing standards but both Broadway and the movies reinforced that culture and fed it endless streams of songs.
Groups were found in a variety of niche cultures. Country & western. Race music. Blues. Jazz. Folk. Early rock ‘n’ roll, R&B, and rockabilly emerged by blending a few of these niches together. You got Bill Haley and Elvis and Jerry Lee Lewis and Chuck Berry and all those other pioneers. The industry was used to promoting individuals, of whichever color, so they were always thought of as a frontman with anonymous backup players. Chess and Motown, the two most influential companies for black musicians, kept this up. Doo-wop groups were somewhat of an exception, but they followed the old path of the harmonizers and didn’t play instruments. When individuals broke out, they stayed individuals, like Dion and Frankie Avalon and Fabian. The industry drove this pattern, and it continued because there wasn’t a good path for any groups to make it otherwise.
That started changing in 1960 when the Ventures and other surf groups made it big as groups. The fact that they only did instrumentals was less of a problem then than now because instrumental records had a long tradition of making the charts. The Beach Boys and the Four Seasons both started in 1962 and both were instant successes. That should have tipped the scales but didn’t. My guess as to why is that there still wasn’t the circuit of small clubs that allowed them to build up their expertise and create a fan base before going national. Look at the brilliant teams of songwriters that are called (wrongly) the Brill Building sound - Goffin-King; Greenwich-Barry; Mann-Weil. They fed their music straight to producers who recorded singers backed by session musicians. The groups existed for radio.
Britain had a completely different history, or at least different in the details. The 1950s started with singers of standards, but without the Broadway/Hollywood homegrown talent stream. It didn’t have the radio culture of dozens of small stations in every city that played niche music. The complete split between blacks and whites in music wasn’t enforced. But it did have a tradition of music halls that led the new generation of entrepreneurs to create a club circuit where being a loud and raucous group was a better draw than an individual crooner.
Natural selection at work. In America, individuals striving for radio play was the better career move. In Britain groups striving for club stardom was.
This world did change after The Beatles. America developed a club circuit; the BBC started playing rock music. You can still find all kinds of places where the two diverged. America’s Sixties ended in huge traumas that the British Sixties avoided, so soothing California Pop and radio pap evolved as a response in one where progressive music emerged in the other. Punk and then New Wave were the response to progressive but the U.S. went into Rap and Disco and Metal because its sets of disaffected youth were black and gay and alienated suburbanites. Grunge wasn’t going to be replicated in Britain, Britpop wasn’t a trend in America.
Huge overlaps occurred, of course. Imitators of all genres abounded on both sides. But I think that looking at the state of music in the 1950s and then the pattern of evolution and response to local variations thereafter explain most of the differences. And there’s a lot of envy on both sides. The British kids bonkers for black bluesmen in the 60s couldn’t understand why we utterly ignored them while American kids who hated Top 40 radio in the 70s wanted the complicated advanced music of the progs and couldn’t get why the punks hated them for being able to play their instruments.
Different environments create different niches filled by different evolution. Happens in music, too.