"I remember a story of a girl in Paradise who ate an apple once. Some wise Sapient gave it to her. Because of it she saw things differently. What had seemed gold coins were dead leaves. Rich clothes were rags of cobweb. And she saw there was a wall around the world, with a locked gate."
- Catherine Fisher
I think there's a case to be made that it was preordained, but not necessarily for the reasons, nor in the way, normally described.
But I also think it could have been avoided, so I'm a bit conflicted on it.
On the preordained side, there's a case to be made that it was necessary for all of Adam's race to suffer the same fate regarding the implementation of death, and that's so that a single act of redemption would be sufficient. Imagine that Adam and Eve were righteous in their adherence to the command not to eat of the tree, that they conceived and bore children, perhaps hundreds, and then two or three or fifty of their children ate of the tree of the knowledge of good and evil. We could consider Adam's grandchildren or great-grandchildren, but it gets messy pretty quick, so we'll stop at children.
So some number of original-sinner children, related through their father, but not to any of the other original-sinners through descendancy, have to be redeemed, separately, it seems. What does it take to redeem them? If Jesus, in our universe, is called "the last Adam" because He was descended from Adam, and therefore can stand in to redeem all of Adam's race, what about Bob's race, Adam's 7th child, with his descendants? What about Ebenezer, Adam's 15th child, and his descendants? Does Jesus have to be born into each family, live a perfect life, die on a tree and rise again two to fifty times. And if He eternally became one of Adam's race in our universe, how could He become one of Bob's race eternally, then also become one of Ebenezer's race eternally, and so on, and so on, in that other universe? How many "only begotten sons" does God have to offer?
Thus, it behooved God to hurry the process along in order to be able to redeem all of mankind, and have the original sin start with the original man. That "hurrying along" appears to be the purpose of the evil Satan, while the good Satan (maybe named Lucifer) had a good purpose in the garden. Like this:
[Psa 91:9-12 KJV] 9 Because thou hast made the LORD, [which is] my refuge, [even] the most High, thy habitation; 10 There shall no evil befall thee, neither shall any plague come nigh thy dwelling. 11 For he shall give his angels charge over thee, to keep thee in all thy ways. 12 They shall bear thee up in [their] hands, lest thou dash thy foot against a stone.
Somewhere the good Lucifer/Satan became prideful and became the evil Satan, who can only pretend to be an angel of light, and then he was available for the devious, but important, work of tempting Adam and Eve.
I imagine that it might be possible that Adam and Eve could have stood strong on the first temptation, and maybe more, but they didn't. Satan didn't leave Jesus alone until he had tempted Him 3 times in the wilderness, and then he still sought other occasions. If Jesus was tempted with some similarity to Adam, then in the same way, God might have allowed multiple temptation events, if necessary with Adam. Adam's fall, then, happening as it did, becomes a sort of comparative blessing, because it allows God to offer salvation to all who are made in His image.
The way Adam's fall would have to be preordained, in order for God not to have to claim authorship for sin, is that either the temptations would have an end (again, like Jesus's tempations), or that they would not be able to conceive children prior to sinning, which seems unrighteous for God, since He told them to be fruitful and multiply while they were still in the garden.
These thoughts are surely not to be taken as any kind of truth, but purely speculation. Yet it makes some sense to me.