I doubt very much that the Beatles (or the Rolling Stones, who stayed true to their blues roots than the Beatles for much longer) lacked confidence in their own accents. They were, as you point out, simply trying to emulate their heroes; musicians have sensitive ears, and some sounds just “go with” some kinds of music. Listen to the way John Fogerty sings the beginning of Proud Mary.

Left a good job in the City
Workin [wəɪkin] for the Man every night and day

Fogerty does not speak that way. He’s on record that [wəɪkin] just came out of his mouth when he was singing – he didn’t plan it, it just happened. It fit the song.

I think it’s worth noting that, except in country-and-western, there was for a very long time a softened or absent rhoticity in most pop singing in America. So I think its a mistake to read too much about vernacular speech into the way songs are sung.

As a matter of interest, the purest Liverpool vowel sound out of the Beatles can be heard in the in the introduction to Do You Want to Know a Secret:

You’ll never know how much I really love you
You’ll never know how much I really care [kɜː]

It is apparently the case that hip-hop is influencing world English. But that’s an entire culture. Hip-hop music may be the original medium, but hip-hop culture is a much greater thing than that.