Understanding the Yom Tov connection of "Kah Keli"

Many Ashkenazi shuls say the piyut of "Kah Keli" (י-ה א-לי) after the haftorah and before Ashrei on Yom Tov mornings. The poem (text available here) consists of three verses, each of which starts with one unique sentence, and then continues with two sentences of refrain. The refrain lists various types of korbanot, and then asks Gd to bring us back to our land.

So what does this poem have to do with Yom Tov?

Looking at the standard Artscroll and Koren commentaries didn't yield anything for me, but here are my thoughts:

1. The opening line of each verse refers to one of the three Regalim:
Verse 1 describes Gd as גואלי - "my redeemer" - Pesach.
Verse 2 describes Gd's unfathomable wisdom - לתבונתו אין חקר - the Torah of Shavuot (granted that Shavuot is mainly about the bikkurim, in liturgy it is closely associated with the presentation of the Torah at Sinai)
Verse 3 speaks of Gd "joining the tents" - חבר כל אהלו - a reference to the Succah. (There may also be a pun in לבלב for לולב.)

2. The refrain is an appropriate introduction for Musaf, as it lists the korbanot and asks Gd to bring us back to our land in the merit of those korbanot.
This is a point that the Artscroll and Koren translations seem to miss, even though it makes the piyut a reiteration of the overall theme of the musaf amidah - remember our korbanot, and bring us back home.
The refrain lists the korbanot of various kinds, and then requests, "זכור נלאה אשר נשאה והשיבה לאדמתך - Remember the exhausted one who lifted up (or carried), and bring her back to Your land." The standard translations assume that this means "the exhausted nation who received Your favour" or "the exhausted nation who bore suffering," because they can't identify an antecedent for what it was that the nation lifted up/carried/bore. But the antecedent is that list of korbanot - "Remember the exhausted one, who lifted up all of these offerings, and bring her back to Your land."

3. And building off of #2, there is another layer of association, which turns Yeshayah's criticism of our festivals on its head.
In Yeshayah 1:14, Gd says that He hates our holidays, they are burdensome to Him, and נלאתי נשוא, "I am exhausted from carrying them."
Our poet reverses the line, describing us as נלאה אשר נשאה, the exhausted one, who carried the korbanot. Gd cannot tell us that He rejects our festivals and is tired of them; we exhausted ourselves, bringing those korbanot to Gd on those festivals, and we claim that merit for ourselves.

So there's a lot a going on here - all of which leads me to Ibn Ezra on Kohelet 5:1, in which he protests loudly against davening with piyutim which require this sort of detective work in order to make sense of them. I tend to agree...

[Added note: I just saw footnote 80 in the Goldschmidt machzor for Succot here, attesting to a late (17th century?) origin for this piyut, and noting its mystical character. Interesting.]