Abram was given the name Abraham by God, and it means father of peoples/nations, not a king. The original king given to Israel was Saul, who was apparently over taken by demons, and had to be replaced by king David, who was a son of Judah, and therefore of Jacob, Isaac, and Abraham The Law and the prophets, the book of the Jews, points to a coming king, who would be king David (Ezekiel 37). As for Aaron, he was never a king, but a high priest. The Levites now represent the priest hood in the Jewish nation, and they are not kings, and are supposedly without inheritance of land.
As for Ishmael, he had 12 sons, as did Edom/Essea, and Jacob, and would be the nations born of Abraham. According to the Jewish prophets, Edom would not fare well in the end. As for the nation of Essau, the Edomites, according to the prophet Obadiah, they shall as stubble devoured by flame. As for the countries surrounding Jerusalem, which would include those who profess Ismael as their father, they shall not fare well if they should raise their hand to Jerusalem, and will be destroyed (Zechariah 12:9). The book of the Jews, given to them by God, according to the Koran, does not look fondly on those who would strike "My people", the sons of Jacob.
Obadiah 1
18 And the house of Jacob shall be a fire, and the house of Joseph a flame, and the house of Esau for stubble, and they shall kindle in them, and devour them; and there shall not be [any] remaining of the house of Esau; for the LORD hath spoken [it].
Kings come from all offspring all over the world in the sense you thinking of. But in the sense of divine kingship and authority, then it's related to how God is King.
Ismail had twelve sons and that was according to the number of princes promised in his offspring, but they weren't princes in themselves, God just made it that their number in sync, and the Kings after Ishmael weren't them, but Isaac, then Jacob, then Joseph, then Benjamin, and the rest of the branches one by one till reaching twelve Successors to Ibrahim.
A prince means inherits a king... so there would be a founder to kings from the offspring of Ismail. This was Mohammad (s), and in Quran we find that before it says Mohammad (s) is found in Torah and Gospels, it emphasizes that authority belongs to God, the world mulk is also used for kingship in Arabic, as well as authority, as well in some contexts kingdom.
We also see emphasis on Twelve after that in Surah 7. So obviously the way to see Mohammad (s) in the Torah is linked to the Twelve branches of Moses who started per Quran with Aaron and ended with Jesus and that the number of successors in Torah is emphasized but with respect to Mohammad (s) while Quran it's emphasized with respect to Musa (a).
The Quran emphasizes in 2:255 that God's seat encompasses everything in the universe, and it's emphasized that God is the true King while Pharaohs while dominant outwardly in the land and apparently have authority, it's God who will account everyone and the consequences of disobeying him are severe while reward of obeying him also great.
In 4:54 God emphasizes that Ahlulbayt of Ibrahim was given the book (Prophethood), Wisdom (Messengership to convey that wisdom), and a great authority (Kingship and leadership, the right to be obeyed from God's right).
It's in this context that we are told we too have those who possess the authority from us (4:59) and they are obviously the envied humans compared to family of Ibrahim in 4:54.
Also there is a disagreement with that sons of Israel means. You said it yourself that Abraham is father of nations, this is metaphoric.
It's same with Bani-Israel per Quran, they are offspring of those carried in the ship of Noah, not even all offspring of Noah.
It's same with our father Adam, it's metaphoric, otherwise, Quran would not emphasized those who God favored his religion to humanity upon and towards and regarding to be of his offspring, because all humans would be of his offspring in this case.
The truth is Adam (a) is the founding spiritual father of humanity, and bani-Adam means we ought to follow a successor of Adam. Israel means the who struggles for God or with God, but there is one such human in all times, that strives in God's way and prays on behalf of people and struggles for the sake of God and fights off the devils and gains power from God.
Bani-Israel thus is metaphorical per Quran.
Also high priest, means exalted leader, exalted is synonymous with being a chosen one, and leader has authority from God to be obeyed and hence is a king from God.
The concept of priesthood in other then chosen ones, was invented and is a wrong interpretation. Aaron is exalted leader, and there would be exalted leaders from his offspring. This doesn't have anything to do with man-made priesthood today.
Ismail is also emphasized to be one who God has heard regarding and who his offspring would be blessed and ways from him would come out and a great pathway from him would come out.