Not trying to bash or anything, so sorry if this comes across that way as it's not my intention.
The potential reasons and ideas of how things are in a recording studio are severely wrong. Microphones are able to pick up quite a bit, especially if it's a condenser microphone. While you can wear headphones, it is extremely common to have the music cranked up in the headphones for the vibe and what not. I guess what I'm saying is that more times than not, they like a loud mix in their headphones.
A click track is usually still in a track regardless of if the rest of the instruments are there.
Additionally, it's a common technique to have one of the headphones on while the other is off so you can hear what your voice or whatever sounds "acoustically" or even how to blend your vocals with other vocalist if you're recording multiple vocals at the same time.
When you mix this all in, it's not hard to believe that a very sensitive microphone would pick up the click track. Try listening to the beginning of Adele's "Rolling in the deep." You can hear the click track there. Oops. Even a song that big is still prone to these mistakes.
The click in OTBOA really bugs me as well. It's definitely from a click track because it has a very distinct sound that you get used to when you use it all the time, so I've gotten used to it. It's very possible that with all the different edits, tracks, time signature changes, crossfades, etc that one tiny area was left un-deleted. You may not even see that it's there unless you zoomed into your DAW/recording program workspace to see it. That really stinks.
It's like most things though, it's abandoned and not completed. Such is art.
I hope this clarifies things a little bit. For another example check out the beginning of Nevermore's "The Godless Endeavor" the title track.