Abinadom sees no need to write much, as the official record is kept with the kings, and Abinadom is not privy to any revelations. As a man of war, he appears to have concentrated on physical survival rather than spiritual survival. The only thing we might glean from his account is that he has a sure knowledge of the existence of the other set of plates, and he knows that they are being maintained. Since those plates are following the reigns of the kings, and the isolation of Jacob's lineage is complete by this time, we must presume that the recording of the history of the people was common knowledge. This would indicate that those plates were somehow referenced publicly, either by reading them or by mentioning the things that are written.
While it appears that the Nephites were a generally literate people from later indications, it would still be consonant with the sacred nature of texts that there be some type of understanding, and perhaps veneration, of the textual history that was being kept. We must remember also that the Lamanites sought to destroy the records, and we may presume that they also knew of them. If the Lamanites know of the records, it comes either from tradition, or the information exchanged during the trading expeditions.
Chronological: Abinadom does not specifically mention passing the plates to his son, and yet his son does have them. When his son Amaleki begins writing, Amaleki is in Zarahemla. It would be very difficult to conceive that Abinadom would have nothing to write had he been on the journey from Nephi to Zarahemla. Since that occurs only in Amaleki's writings, we must assume that Abinadom lived out his days in Zarahemla.
Perhaps this "oversight" in delivering the plates to a named successor is not necessarily an oversight, but a recognition of the fact that Amaleki was very young at the time, and could not receive them directly from his father.