SoP, you make a few good points, although I don't see quite how the account of the creation in Genesis can be ascribed to support the birth of the Israelite nation, as this really begins with Abraham, who appears to show up from nowhere in the account of Genesis, completely dissociated from the previous accounts.
Eden
One must realize that the story of Adam and Eve, in fact, all the pre-Abrahamic mythology is Jewish in nature. It is an explanation of the singular importance of the Jew/Israelite in an existing world. Per the creation narrative, only the progenitors of the nation, i.e. Adam and Eve, were blessed to be in the garden, directly created by the Almighty with a singular relationship. His presence is felt by them in the Garden. They are special.
The story moves from there. The legend builds. A legend or myth structure shared over the centuries, from Adam to Abraham then to Moses, its supposed scribe. To say that the story is "completely disassociated from the previous accounts" is to reject the lineage from Adam to Abraham that is contained in the Torah itself. It is a package deal. All related. Adam to Abraham to Moses to NATION.
Take as an example the story of Pocahontas and Captain John Smith. The story (legend) of the first settlers (colonists). The story, although it is displaced from the Founding Fathers legend, is still repeated in history books as part of American History. It serves to distinguish the "american" as a trail blazer, confronting hardship, worthy of help from the indigenous. It also serves to promote the notion that the early settler was justified in displacing the indigenous. After all, Pocahontas marries John Rolfe and converts to christianity, if it was good enough for her (legend) then history would expect that her people would follow suit. It is all part of the folklore and legend ritual necessary in the foundation of a nation.
Hope that answers your question.
SOP