Something I find interesting in Evangelion is how the repetetive monster of the week thing even if it is sometimes boring can enhance other effects. The release in the scene where a certain symphony by Beethoven is played is made much more powerful by it being the point where the pattern ends or comes to completion.
The repetition, be it from
Eva, or
Utena, or
Star Driver, can be frustrating to some people who look to the action sequences for the conflict, but the action is really just a backdrop to the internal conflict of the characters and their development, which is where the real narrative lies.
You bring up music, and I think the structure of these kinds of narratives can be very well described by the structure of a classical fugue, in which a "subject" is stated, followed by various musical episodes and development of the subject in various keys, before finally returning to the opening key, and a coda.
To relate this back to
Eva, we have the introduction of the angels, and episodic nature of fighting them. This motif gets developed and changes, but is truly only there to underscore the development of the characters. Reflecting this, as the characters are cast further and further into their depression and disconnection with reality, the angels — which began as fairly humanoid with Ramiel — grow less and less humanoid until we get beings like Leliel, which doesn't exist in our dimension, Arael which attacks by "raping" Asuka's mind, and Armasael which has no form and tries to absorb the Evas. Many of the battles grow less and less action and fighting oriented, and become battles that take place inside the character's minds, particularly noticeable when Leliel absorbs Shinji and when Arael's mind-rape attack ultimately reduces Asuka to fugue (hah) state. Before finally, we have a return to the original form, the original key, with Tabris, or Kaworu, who comes in the form of a human, and a coda in the ending.
In the same way a fugue plays with its musical subject and develops it, a show like Eva develops the motif it sets up in its fights and changes it to reflect the true nature of the narrative, this is, its characters. As the characters break down, so does this motif, as you mentioned, and Anno goes even further, as ultimately the very concept of traditional narrative is altogether abandoned. When the repetition breaks down, so does Shinji, as well as the narrative and the narrative's concept of "reality" itself, with the final two episodes, and the second half of
End of Evangelion taking place in a meta-reality inside the minds of the characters. In this disconnected place, we have the coda that takes us to our ultimate conclusion, which is a return to reality, in a new state, i.e., a departure from the "fugue state" (heh) in which the characters — and perhaps ourselves — had fallen.
Speaking of music and
Eva, anyone who does not break down when "Komm, süsser Tod" begins to play in
End of Eva does not have a heart. Talk about release.
ETA: Likewise, though every episode ends with "Fly me to the moon (In other words)," every episode uses a different arrangement of it, often sung by the different seiyuus.