Each set of four has a song title and a group or artist title, and the clue mentions that the song is by a different group than the one mentioned. For example, Path or Glory is by Heaven's Gate, which is not mentioned in the set. But the next song mentioned is Better Off Dead, which is by New Found Glory, the group mentioned in the first clue.
So the songs and group form a chain: each song has one word (or subword) in common with the group in its clue, but it's actually by the group in the next clue, looping around at the end. We conclude that the first blank is a song by Dire Straits that has one word in common with "Heaven's Gate". "Ticket to Heaven" is the only one that matches.
We can complete the blanks in the three sets as follows.
First set:
Second set:
Third set:
We can now iterate this procedure on each of the three sets above, to produce the following set.
Repeating the procedure one final time on this set, we get the final clue: Working for the Weekend, not by Vampire Weekend.
The answer is the common word, WEEKEND.