On a Conjecture: "The God of X will be the Son of X"

First, that the premise that "The God of X will be the Son of X" is incorrect and/or has some exceptions beyond Elijah, can be established. The statement that: "Going through the ESV, I checked every verse with the construction 'God of X'. When X is the name of the person, these are the only names ever used" is false.

Nahor

The God of Abraham, and the God of Nahor, the God of their father, judge betwixt us. And Jacob sware by the fear of his father Isaac. (Genesis 31:3, KJV)

Nahor was Abraham's brother:

And Terah lived seventy years, and begat Abram, Nahor, and Haran. (Genesis 11:26, KJV)

Nahor was also Abraham's grandfather:

And Nahor lived after he begat Terah an hundred and nineteen years, and begat sons and daughters. (Genesis 11:25, KJV)

The phrasing "God of Abraham, and the God of Nahor, the God of their father," may well refer to the father of Abraham and Nahor, i.e. Terah--but Nahor himself would then be a separate line from that of Christ. If this does not refer to Terah, who "their father" is remains quite vague, as this has already made a generational jump, and could quite well be expected to do so again. The most logical application seems to be that "their father" refers to the mutual father of Abraham and Nahor, which was Terah (and Haran may not be mentioned on account of having died early, even before his father).

The fact that "God of Nahor" was not included in that original list may indicate that whomever compiled it was not comfortable with it fitting the desired paradigm, therefore omitted it. However, the list is not then comprehensive, as was claimed.

Jeshurun

Only three verses in the KJV translation name Jeshurun (shown below).

But Jeshurun waxed fat, and kicked: thou art waxen fat, thou art grown thick, thou art covered with fatness; then he forsook God which made him, and lightly esteemed the Rock of his salvation. (Deuteronomy 32:14, KJV)

And he was king in Jeshurun, when the heads of the people and the tribes of Israel were gathered together. (Deuteronomy 33:5, KJV)

There is none like unto the God of Jeshurun, who rideth upon the heaven in thy help, and in his excellency on the sky. (Deuteronomy 33:26, KJV)

The second verse of these shows Jeshurun to have been the name of a place. However, it is common for people's names to have become names of places, whether of cities (e.g. "city of David", "Dan", etc.) or countries (e.g. "land of Israel", "Moab", etc.), so this does not establish that it was only a place. Looking closely at the first mention of Jeshurun, we find clear reference to him in the third-person masculine singular in Hebrew. This is unlike cities, which are always grammatically feminine in Hebrew, for example "Dan" is a feminine-gendered noun in the following verse (which also shows it was the name of a city).

And when Abram heard that his brother was taken captive, he armed his trained servants, born in his own house, three hundred and eighteen, and pursued them unto Dan [N‑proper‑fs]. (Genesis 14:14, KJV)

So having Jeshurun addressed as "he" and "him" indicates this was a man's name. Arguing that "God of Jeshurun" only referenced the place of that name, and not the person of that name, would be a significant stretch, indicating subjective bias.

Jesus

That the God of our Lord Jesus Christ, the Father of glory, may give unto you the spirit of wisdom and revelation in the knowledge of him: (Ephesians 1:17, KJV)

The premise of the list-maker seems to be that if an uninspired/foreign person (non-prophet/non-Israelite) had used the expression, it was not to be counted. But here we have the inspired apostle Paul, a Jew, speaking of the "God of our Lord Jesus Christ."

This verse has the opposite effect of that concluded in the OP. In place of saying "The God of X will be the Son of X," it indicates that such a construction is invalid, because God is not the son of Jesus.

Conclusion

The premise is invalid. The original list compiler appears to have, whether mistakenly or deliberately, omitted several key occurrences of this "God of X" expression throughout the Bible--coincidentally the ones that would invalidate the premise. It may even be that the compiler chose the ESV to avoid Deuteronomy 33:26, which the KJV has as "God of Jeshurun," but which reads "God, O Jeshurun" in the ESV. However, the other two among these exceptions still exist in the ESV as well, rendering its "every verse" statement false, even with the ESV.