I think it is neither controversial nor over-the-top to say the Kline is an enemy of a "literalist interpretation of the Genesis creation 'week'" (his own words and characterization).
It is just naive and silly to suggest there is not first and foremost in Kline's framework hypothesis a determination to understand the Bible in a way that produces an interpretation (hermeneutics) that is not in conflict with anything we "know," humanly speaking, about the world.
"Framework" purports to take the biblical message (in Gen1-2) and separate it from the idea that data points in the Bible correspond to natural history. If these two may be distinguished, then free-inquiry can be made, and neutral observations made about the world, in pursuit of "truth". There are no ultimate standards of accuracy in the natural realm.
Logically, there is no reason to assert that this hermeneutical proposal has to stop at the end of Gen1, or Gen2, or Gen3, or Gen12, or Gen50, or Ex15, or... These are arbitrary stopping points; whatever criteria are chosen to designate a place where the Bible takes up "real-world" history are not themselves subject to the same principles of revision.
Therefore, for example, if an archaeologist finds a "king list" of the Davidic monarchy, and it appears to create a whole new timeline--including for instance reversals of "traditional/biblical" father-son relations, and new names, and even the end of David's physical lineage--then if the weight of "unbiased" scholarly opinion takes the side of the inscription, the biblical data may be given "theological" priority, but its "history" should be subordinated to the experts.
There is no principled reason why this process should not take place if the philosophy of the Framework hypothesis is accepted as legitimate. To suspend the principle at the end of Gen2 is purely arbitrary, and serves a present-day theological interest of conservative-credential.
The question can be reversed. Given the reality of FH, what principled objection could EVER be raised, that would demand a biblical chronology or king-list or historic claim trump the "findings" of archaeology, geology, biology, etc.? There is none. And for this reason alone, FH should be relegated to the scrap-heap of hermeneutical history.