So, my question is a question of mechanics.
Once Shub-Niggurath spawns as an Epic Monster, it specifically states that it cannot TAKE DAMAGE (rather than 'be defeated', for example) until 3 mysteries have been solved. Does the Flute of the Outer Gods (which automatically defeats all monsters on your space provided you pay the cost of 2 sanity and 2 health) work in killing him before the 3rd mystery has been solved?

I don't have the game with me at the moment, so I can't really check, but providing the wording you reported is accurate, I'd say that yes, it's legit to use the Flute, until no "damage" is mentioned, but "defeat"

I don't have the rules in front of me and have never played this scenario yet but I believe there is a rule concerning Epic Monsters that says they may only be killed by damage being caused to them. I interpreted this as any effect that causes damage.

however the Flute would kill everything else...then you could start hurting Shub-Niggurath.

Aye, correct. Reference guide, pag 6:

An Epic Monster cannot be defeated by any effect except losing Health equal to or greater than its toughness.

Yep. Any effect that moves or defeats a monster without can't affect epic monsters and thus can't affect Shubbie. Stuff that damages a monster like Dynamite, Agency Quarantine, Shriveling, certain Tokyo encounters or the like can hurt her if you have thee Mysteries solved.

I will never beat Shub-Niggurath. Oiy.

Beautiful. Okay. When the situation came up, I assumed that would be the case because that would be absurd otherwise.
However, in a single player game, I pulled a Tokyo encounter that caused me to be delayed and dish out 3 damage to any target on the map, when Nub was up (and had 3 toughness). That event won the game for me. Classic.