The Jewish understanding is that the story could not be about passing on death to all because the narrative of Adam and Eve ends with their being blocked from partaking of the Tree of Life.
In other words, the story implies that if they ate from the Tree of Life they would live forever. Since that was required for eternal life but Adam and Eve never got the chance to do that, therefore they never had the gift of unending life to begin with. You don't need a Tree of Life or its fruit if you already can live forever. So it could not be eternal life that Adam and Eve lost "due to sin" because they never had access to it to begin with. They were obviously created as mortals, and death was part of the plan of G-d from the beginning.
But this is not the meaning of the narrative however. Recall that this is the second creation story. In the first, in Genesis chapter 1, there is no mention of trees with fruit that gave knowledge or life. Instead in the first creation story no foods are off limits.--Compare Genesis 1.29 with 3.3.
In the second creation story (Genesis chapters 2 and 3) we are introduced to our first parents and an "incident" which seems to teach an axiom: humans make mistakes, and that's part of our nature. The greatest mistake is leaving G-d out of the equation or decisions we make. "Sin," which is never mentioned in reference to Adam and Eve, doesn't even show up until Cain starts thinking murderous thoughts. (The first mention of sin in the Bible is connected with fratricide at Genesis 4.7.) The story is not history but a creation narrative, setting the stage for the third creation story, that of the Noachin flood (Genesis 6-9).
The third story employs a more common creation narrative accepted among the ancients, one that stated that humans and the world have always existed but that the current world is the result of the other being wiped out by a flood (floods of water were seen as creative not necessarily destructive forces). Both evil and the "curse upon the ground" created due to Adam's disobedience (Genesis 3.17-19) is reversed in the third creation-flood narrative. The name "Noah" means "the one who relieves us of the curse."--Genesis 5.28-29; compare the fact that after the flood, Noah is able to make the earth produce crops, Genesis 9.20-21.
All three stories are but one, describing the same "miracle" of creation, but employing three separate but accepted cosmogonies of the ancient Mesopotamian world. While Jewish tradition holds that Adam and Eve and Noah were real persons, it does not imply that the narratives about them are historical. The middle creation story says that Adam brought a curse into the world, one that Noah relieved us of through his surviving of the flood.
With the curse lifted creation could now echo the words originally spoken of in the first creation story, that the world and humankind were essentially created as "good," not sinful. As such people could be holy by being obedient to G-d, unlike acting as Adam and Eve once did. These stories are merely setting the stage for the main player, the arrival of Torah itself which is the means by which Jews believe holiness can be introduced into the world both directly and indirectly.--Leviticus 11.44; 19.2.
The Torah itself, however, is not the only means to holiness. All humankind has this capacity built in it. The Torah is the means offered to the Jews, but when other nations follow their conscience in ways of justice and right, they bring about holiness in them and the world around them too. This is the meaning behind the creative narratives, that G-d gave us the capacity to be as G-d declared us from the very beginning:
"God looked at everything God had made, and God found that it was all very good indeed."--Genesis 1.31.