None of us go around saying "I'm listening to a song by Eagles." Whether or not "the" is officially part of the band name is irrelevant; they picked a name that is also a common noun, and therefore grammatically, it sounds unnatural to omit the article in the sentence when using it. Glenn himself uses the article the vast majority of the time for this reason - just watch the very interview this thread is about to see that. I'm sure VA is right in that there have been occasions where he hasn't; I imagine those were occasions where it sounded natural grammatically to omit the article. Those occasions are rare, however, in the English language.
I just don't capitalize "the" in order to acknowledge the fact that the band's name is officially "Eagles" without the article, since this issue is apparently important enough to them to make it onto a tshirt!
Funnily enough, the two photographers who did the Eagles' first two photo shoots (Diltz and Elder) had this same conversation on the "Under the Covers" documentary 20 years ago. This has been a matter of great concern for a while.
Now, about the interview, when Glenn says you have to write every day if you're going to write songs, do you think he means that's what he's doing now, or do you think he means he can't do that now? I hope it's the former!