There doesn’t seem to be any consistency in usage. Why does it seem normal to say that John Lennon was a Beatle, Ray Davies was a Kink, and Don Henley was an Eagle when it seems wrong to say that Jim Morrison was a Door or Ric Ocasek was a Car?
I don’t know that I would say Ray Davies was a Kink. He was a member of the Kinks, more in line with your Morrison and Ocasek examples. But I agree there seems to be no clearcut, systematic formula. Bands named after animals seem to follow the Beatles/Animals/Eagles rules. But what about Los Lobos, which translates to The Wolves? Is David Hidalgo a Lobo or part of Los Lobos? I’d say here in the US, he is part of Los Lobos, not a Lobo. What about in Mexico?
Because “door” and “car” are everyday words, and require a certain amount of disambiguation.
I think it’s a continuum. See my post #4 above - there is tendency to prepend “the” when a plural band name could plausibly (in some poorly defined sense) refer to the band members. How we refer to individual band members also seems to follow this loose rule, with clear examples at each end of the spectrum but a large grey area.
At the extremes:
the Dixie Chicks, each member is definitely a Dixie Chick
the Spice Girls, each member is definitely a Spice Girl
10,000 Maniacs - there are not 10,000 band members, and they are not Maniacs
Dire Straits - obviously not, it’s a concept rather than a description of the members
Simple Minds - presumably a concept
In the grey area:
a Beatle? - sure sounds right, even though a “beatle” is not otherwise a thing
a Kink? - dunno
If the construction is X and the Y’s, I think maybe I’d be more included to call the members “a Y”, even if it’s a little silly - a Pip, a Blowfish, a Supreme. But I guess not a “New”. Is there a story behind the name “Huey Lewis and the News”?
And then there’s The Mamas & The Papas. Their album covers, at least, tend to capitalize both The’s. Cass Elliot was frequently called “Mama Cass,” but nobody ever talks about Mama Michelle or Papa Denny. There is, of course, a Papa John, but he makes pizza. 
And despite her being called Mama Cass, I would never say, “Cass Elliot was a Mama.” I definitely wouldn’t say "Cass Elliot was a Mama & a Papa.
As for XTC, INXS, R.E.M. and the like…Neil Young came up with a solution one day around 1987 when he saw Peter Buck in a diner and said:
“Hey, ain’t you one of them rem boys?”
True story.
I’m going to a Who concert.
Cool. Who are you going to see?
Who.
That’s what I’m asking you. You’re going to see Who?
Yes.
They’re not in town until next week…
I can’t find a better quality recording of this old SCTV sketch, but it’s pretty darned funny.
I was going to say that on Ringo’s bass drum it said “The Beatles” but I just checked. Both “THE” and “BEATLES” were capitalized, but the B and T in “BEATLES” were larger.
The Sweet as well.
For those doubting there could be an individual “Kink,” their guitarist Dave Davies wrote an autobiography called Kink, which, to be sure, is a double entendre to some extent.
I remember an interview Billy Corgan gave (on MTV?) where he explained that “smashing” was an adjective, not a verb, as in British English where you might say, “That’s simply smashing!” As such the band name is a pun also alluding to pranksters smashing pumpkins around Halloween, but doesn’t explicitly refer to that act. As a guess, they added the “the” later on to clarify.
It’s also possible that they retroactively made that up and it wasn’t originally a pun, I wouldn’t put it past Billy and/or the band’s publicists.