Then what has the body to do with the soul in the matter of transferring Adam's sin?
The body provides the direct link back to Adam! If all human souls weren't created with Adam or immediately at the moment of Adam's sin, then how could they inherit his sin? Thus, if souls are created later on, the only connection we have to Adam is through our bodies, unless God goes out of His way to implant original sin into each newly-minted soul, like a factory purposefully releasing products with defects (by, say, denting each new car with a hammer as it leaves the factory), which would defeat the whole point... souls are perfect when created... what is flawed is the body, which passes on its defect to the passenger soul.
Adam and Eve.... from them are all human beings bodily descended... meaning the flesh and blood of human beings, their bloodline, can all be traced back to a primordial first man and first woman... but with this ancestry comes the heritage of original sin... original sin is passed on through the body... through the flesh... through the blood... maybe, in contemporary terms, genetically coded!
Thousands of years later... generations and generations of parents and children later... let's take a random couple... at the moment of conception of the fetus, the fetus is endowed with a unique, pure, immortal soul (with distinct beginning but no end)... as a result of the commingling or joining of body and soul, the Adamic sin rubs off on the soul and the child, when born, must be baptized in ritual cleansing... but it's only ritual... throughout its life, the child, the new body-soul being must accept Christ, ratify Christ's covenant in his/her heart, and be judged worthy of entering heaven as a shriven soul post corpore-partem.