No, I do not believe the sin passed on from Adam is “bodiless” On the other hand, I believe the sin of Adam instigated by Satan became a sin capacity that was passed on from individual to individual by physical birth. This is the doctrine of the federal headship of Adam (in Adam versus in Christ). As “in Christ” is a spiritual status so is “in Adam” a spiritual status.
To assert that the “seed” is a physical seed produced by Satan reads into the text something that is not there or anywhere else in the Bible. This is the hermeneutical error called “interpolation.” The onus of proof of such an assertion falls on the person who asserts the interpretation.
Everything in the New Testament dealing with the sin capacity (“flesh,” “the Old man,” “seed,” “sin” in the singular versus sins in the plural for example) refers to the fall of Adam, not some supposed sex act. This is the argument of Romans 5-7 and well as numerous other passages. The history of the church argues this point, the history of biblical interpretation argues this point. The difference has been between federal headship of Adam versus the traducian view of the nature of man. The body carries the sin capacity–”As in Adam all die, even so in Christ shall all be made alive.” “In Christ” was not physical and the essence of “in Adam” is sin, not physical.
In other words, my issue is not “shock” but a demanding exegesis, exposition of Scripture that will not allow eisegesis or reading into the text something that is not there. This is especially true when we consider the overwhelming evidence of how the New Testament looks at this issue. The New Testament is didactic, not narrative, which gives greater specificity to the issue.

It appears that the basic reasoning behind your insistence on a bodiless progeny for the serpent is that for his progeny to have a physical dimension there had to be sexual intercourse between him and Eve. In that respect your thinking matches that of the Branhamites and their followers (including Andy) who believe that “physical progeny = need for sexual intercourse”. I understand your shock at and rejection of a doctrine which, to you, implies an unimaginable sexual act. There are some parallels here with Nicodemus throwing his hands up in horror in Jn3.4. The mistake in both cases (although Nicodemus may be intentionally exaggerating, which I do not think you are) is over reliance on simplistic human reasoning which sees a need for human sexual involvement to viabilize an act of God. If a born again believer can be generated outside of human sexual involvement why cannot the serpent’s progeny be generated outside of demonic sexual involvement? If the former can be a flesh and blood being why must the latter be a disembodied spirit?

Please read this and double-check if needed. From the Jewish Encyclopedia, Vol. XI (copyright 1905, page 69-70): “Satan was the seducer and the paramour of Eve, and was hurled from heaven together with other angels because of his iniquity (Slavonic Book of Enoch)…The chief functions of Satan are, as already noted, those of temptation, accusation, and punishment. He was an active agent in the fall of man…and was the father of Cain…The serpent of Gen. iii. is identified with Satan.”