The usage of the capitalized definite article “the” on the band names is one of the worst pedantic stupid arguments on the straight dope.
If the name of the band is Who (no “the”) then you don’t say “I like the Who”, you say “I like Who”. And then everyone says “Who?” and you realize how stupid you sound.
Tell me, what do you do with The The?
Pssst. Just above you.
That wasn’t there when I was typing!
The Name of This Band Is Talking Heads.
Oh, come on. We’ve had much worse pedantic, stupid arguments. We’re barely going half speed (or is it half-speed?) here.
That’s nice but trademarks have nothing to do with how to write “the” mid-sentence, whether it’s a band or a football team or a magazine, all frequently named with that word at the head, and yes that word is part of their name which is irrelevant; the convention is what I (and style guides to written English) have said,.Conventions may evolve, but as of now the Band and the Who will just have to live with it.
Nice dodge on The The. So how should The The be capitalized?
Trick question: is it the Eagles, or The Eagles?
A name is a name, and that’s how you write it. The rules of grammar don’t decide whether Leonardo is da Vinci or Da Vinci, or whether “The” is or isn’t a part of Ohio State University.
Are we having this argument again?
You don’t say:
the Pink Floyd
the Led Zeppelin
the King Crimson
the Menudo
the Blood, Sweat and Tears (whether the Oxford comma is there is open for debate )
You do say:
The Beatles
The Rolling Stones
The Eagles
The Doors
The Who
The The
I am agnostic if you can or should say The Talking Heads.
Well, they made it crystal clear with the title of their live album.
eh, what do they know?
This, on the other hand:
You beat me to posting this iconic poster. It’s hanging on my living room wall. 
I wore this t-shirt (without the specific place/date) in high school rather a lot. Great picture.
It’s just Eagles. No “the”.
The Dude says otherwise: “I hate the fucking Eagles, man.”
I mean, the (or The) Dude’s not dumb, he knows if he said “I hate fucking Eagles” they’d all think he was unAmerican.
Clearly, “the The The.”
aka “Die Bart Die” in German.
Or into bestiality. “Thiose buggars are hard to hold on to. And watch out for the talons.”