An "orthodox" theologian would probably object that you are mixing up the transmission of "sin" (in the very basic sense that every human being "is a sinner") with the fact of being punished for somebody else's "sin" (as a specific offence). In the classical "original sin" doctrine everyone dies for his/her own sins -- just, being "sinless" isn't an option.
There is a basic contradiction in the concept of "sin" which I think can be traced back to its dual origin in ritual and moral/civil law. The earliest use of the Hebrew words usually translated as "sin" belong to the priestly vocabulary: here "sin" has strictly nothing to do with morality and everything to do with ceremonial uncleanness: unwittingly touching a corpse, getting a skin disease, moisture on the walls of your house, giving birth, having sex within "marriage," are all "sins" barring you from contact with the sacred until you perform the appropriate rituals. This kind of "sin" is highly "infectious" -- it communicates through mere contact -- but it is not a moral offence (only refusing to go through the purification process would be).
Then prophetic texts have (in an initially metaphorical and subversive way) extended this priestly vocabulary out of its original scope to apply it to moral offences such as dealt in the civil law (murder, adultery, theft, etc.), and beyond. The use of "sin" in the Torah reflects both influences. It should be noted, btw, that it took some time before the principle of individual punishment (for "sins" in the sense of civil or religious offences, not ritual uncleanness) was carried over from the realm of human justice to that of divine justice. Deuteronomy which limits human punishment to the individual wrongdoer still accepts that Yhwh punishes the sins of the fathers on the children to the third of fourth generation. Only later (cf. Ezekiel 18) did the "dissonance" kick in.
The later Christian (and rabbinical) concepts of sin depend on this complex and confused heritage. It cannot think of "sin" in a purely ritual (non-moral) manner as the older priests did. The concept of moral flaw or failure is always attached to "sin". But on the other hand the old priestly concept persists in the idea that "sin" is somehow "infectious," and exceeds individual actions and retribution. This may shed some light on the Augustinian idea of "original sin", as a new rationalisation trying to make sense of this basically contradictory concept of "sin" (in a sense which is not strictly "scriptural") as both moral and inherited.
