The Hebrew term for "high priest" is often not present in the Samuel-Kings narrative. Instead there's often just "the priest" with a context making it clear that this is a priest who is over the other priests. Sometimes the Torah passages that discuss the high priest are therefore thought to be a late addition (the implication being that the Pentatechal narratives are deceptive in presenting a false history to justify a later practice, or else some view gets concocted according to which everyone knew it was presenting a fictional history of Israel so that it only deceived later generations).
I have little patience with speculative reconstructions of the history of Israel that don't actually have evidence supporting them, especially when the actual evidence we have is only what we see in the actual historical narratives we have, narratives that certainly present things in the order of Exodus coming before Kings and not the other way around. The onus of proving an alternative hypothesis seems to me to require a bit more than things like the absence of terms like "high priest" in the writings discussing later periods. We'd have to have no available explanation for why the term might have dropped out for that to count as much evidence.
Nevertheless, the fact that the term "high priest" is hardly ever used in the Samuel-Kings narrative is at least a very small piece of evidence, even if I wouldn't consider it enough to outweigh the strong presumption of the only chronology we're ever given. It's still something that ought to have an explanation, and it's worth thinking through what that might be. The usual term is "the priest" in Samuel and Kings, and that isn't what you'd expect given the Pentateuchal narratives that precede Samuel-Kings in the traditional chronology.
There seem to have been two figures taking that title during the time of David, and there's no statement that they were back-to-back. They seem to have been simultaneous. I've explained in another post why I disagree with the speculative reconstruction about Zadok that holds sway among most liberal scholars and why I think a better reconstruction of events is that the high priesthood changed hands from the Eleazarites (descended from Aaron's third son) toward the Ithamarites (descended from Aaron's fourth son). That seems to have ended when the line of Eli finally died out. From that point on, there was one priest called "the priest" among the other priests, and it was always a Zadokite from Eleazar's branch of the priests. The only geneological information we actually have traces Zadok to the line of Aaron's third son Eleazar and Eleazar's son Phinehas. (The oldest two were killed in Leviticus during their early days as priests, and presumably they had no offspring.)
But why the terminology of "the priest" rather than the Torah term "high priest" if the Kings narratives were indeed written after the Torah texts that discuss the high priest? Remember that the period of the judges led to such decentralization, with every priest doing what's right in his own eyes, that there was in effect no one high priest but just people taking that position in particular locations. Perhaps Eli was one of them. You might have had relatively faithful descendants of Aaron here and there, and the evidence suggests that other descendants of Levi were functioning as priests also, but there's no reason to think very many of these priests or Levites were faithful and much reason to think many weren't. It's quite possible that Eli was the highest geneologically-descended priest who was faithful in his time, and thus he was called high priest because he was the most rightful high priest willing to serve the proper function of a high priest. If the entire line of Eleazar was unfaithful during this period, we might expect something like that. But when someone faithful to God's law like David came along and began to rely on Abiathar, a descendant of Ithamar, as his sole priest, he might have been loath to call him the high priest because the high priest should be descended from Eleazar. Then when he found Zadok, who had every reason to be called the high priest, he was unwilling to demote his close friend and confidant Abiathar, and they were thus both listed as "the priest" in parallel for David's early reign. Eventually, Abiathar betrayed David and sided with Absalom, and that ended. From that point on, Zadok was "the priest". I wonder if terminology just stuck, at least in some quarters, and you didn't get a resumption of high priestly terminology until a later revival reintroduced the scriptural term.
That reconstruction of events makes a good deal of historical sense, given the text's presentation of chronology (which, again, is our only strong evidence of chronology). Scholars pick at little things that are hard to make sense of without some thought about what might have happened, such as the sudden appearance of the Zadokites and the use of the term "the priest" instead of the term "the high priest". But it seems to me that there are explanations available for these things, and the speculative reordering of events to suit some contemporary fad at undermining the historical accuracy of the biblical narratives seems less intellectually-motivated to me than does the simple effort to find a way to make sense of the historical texts we've got, which does take some speculation but not on such a grand scale. So even without any motivation to maintain the historical reliability of scripture, I think there are good intellectual reasons to resist the revisionist readings.