I agree with the second part of what you said re: Odin and the older beings. However I disagree with Cowl not having obligations after claiming the power of the Darkhallow.
If Cowl were to claim the Erlking's throne then he'd be bound by the Erlking's obligations (though after eating Chicago GodCowl may be strong enough to get out of them). If the Darkhallow doesn't eat the Erlking in the process and GodCowl becomes a god of Necromancy instead I could easily see them having obligations to other death gods after stealing the harvest so to speak.
My personal theory on the Erlking's purpose was a ploy by Odin to eliminate the nastiest of the old forest spirits and lock the power into the cycle of the courts. This is why we don't see references to Herne the Hunter despite the Wild Hunt running around. Humanity gets safer without the old nasties and Odin gets an ally who owes him his power. Win win.
I don't believe the Darkhallow would have consumed the Erlking. That's not it's job. The Erlking was summoned so that he could raise those old, powerful hunter spirits. Those spirits have much more metaphysical mass and when consumed in the Darkhallow would have allowed the recipient to become a much stronger god than they would have otherwise.
So I don't see how if Cowl had completed the Darkhallow he would have taken up the Erlking's obligations as the Erlking would have still been around to look after them.
I also don't see what obligations Cowl would have had to take on when becoming a god. From what I can see, the power of the Darkhallow is black magic based fuelled by powerful spirits. But what debt would have Cowl incurred? What responsibilities would he have taken up? I think the Darkhallow was the cheats method of becoming a god, and whoever completed it got the power without the responsibility. I could be wrong though.
Just FYI in Cold Days, Eldest Gruff calls the Erlking "Lord Herne" during Harry's birthday celebration at Arctis Tor. It's the only reference to it but it is there.
He's Summer King, per WoJ- Kringle is Winter King. Kringle embodies the spirit of giving in Winter but is still Winter, Erlking apparently some form of "red in tooth in claw" or the joy of the hunt.
They're both Wyldfae, but associated with those courts most strongly.
Jim has contradicted himself on this before. Initially that was what Jim said but later he also said the Erlking has Winter origins or something to that effect.
As you point out though, they are both Wyld Fae which means they can shift around a bit.
Did Jim mean:
1) When the Fae were "Called" to their Courts, what became the Guard went to Harry, so he stuffed them in his bag? This would mean they recognized him as their "court."
2) That the bag held box cutters, and Harry summoned them by Calling into the bag to attack Aurora?
As regards the big E: I agree his purpose- aside from referencing Labyrinth and the late great Bowie- is to provide focus for the goblin's mischief.
I think he meant the first one myself. But I can see how people might think the second one is true.
Yes, definitely a Labyrinth reference there.