Actually, don't know why any of the world's kingdoms stuck with the Pope. Orthodoxy is just so much better from the state's point of view
Orthodoxy is better because:
The Patriarch isn't the Pope, makes no claims to universal head ship of the Church, allows you (actually requires you) to approve all Bishops in your territory and believes that the Church is subordinate to secular law that doesn't conflict with scripture. The head of the Orthodox Church is the bishop of the local diocese subject to the discipline of the local Synod so all religious matters would be handled right at home All nice things for a Prince. Control over Church lands is the biggest struggle of the Reformation- in Russia, it was a bookkeeping entry
This still doesn't deal with the problem of "but they didn't go orthodox".
Why did the Merovingians and Carolingians give the church so much land and provide so much deference to the Bishop of Rome? I know being Trinitarian rather than Arian gave them a big political advantage over Arian enemies in gaining loyalty over local populations of laity and clergy, but couldn't keeping mastery over the church like the eastern emperors have worked about as well to gain that edge?
Now once the Merovingians and Carolingians had proceeded well along the path of deference to the Pope, I suppose that it was not really possible to get out of that rut given how much property the church acquired and the ability of increasingly independent nobles as we moved into feudalism to play Emperor and Pope against each other.
So, I guess that accounts for developments in northern Italy, the HRE, Croatia, France, the Low Countries and northern Spain.
But why wouldn't other Kingdoms that became Christian before the formal split of 1071 (England, Asturias, Benevento-south Italy, Denmark, Norway, Sweden, Poland, & Hungary) have kept the church under their thumb and autocephalus in the style of the Byzantine Emperors (and Bulgarian, Serbian, Georgian rulers)?