he was promoted for time served without ever having to pass any tests or board.
He deserted an E-1 private and was promoted for time in rank.
No way was he only an E-1. You make E-2 automatically upon completion of basic/AIT, before assignment to a unit. If he had some time in the unit before it went overseas, that would have been how he got to PFC (E-3) before going out to find himself, i.e. deserting. The other advancements in rank came from the operation of time, as you say. With the classification as POW, that was SOP and has been since at least WW2.