Trudgill noted the American-ness got less aggressive as time wore on, and the British Invasion acts became more comfortable with their native speaking voices. By the time the '70s arrived, punk bands like The Clash were turning away from American affectations. Hard.
That's one explanation, but seeing as we still have modern fakers like Adele, it's incomplete. Some people argue that the phenomenon is more a matter of technique. Billy Bragg, who's normally pretty cool with singing like a Brit, once said, "You can't sing something like 'Tracks of My Tears' in a London accent… the cadences are all wrong." (Lest you doubt him, here's Bragg singing that Smokey song.) A recent study by Andy Gibson, a sociologist in New Zealand, would appear to back Bragg up. Gibson found that Kiwis defaulted to an American singing voice across the board, and it wasn't a conscious choice. He surmised it was just easier to sing in that accent. That's partially because of the way we round off certain words when we sing, and partially because the world is so used to hearing American accents in pop songs, it requires more effort and concentration to sing in a different accent. Even if that "different accent" is your default speaking voice.