When DS received his waiver from USMA he reached out to his USNA admissions officer and asked about the waiver process. USNA admissions officer referred him to USNA medical personnel. When DS asked about the waiver process he was told "when you are in line for an offer of appointment to USNA or one of our preparatory programs, we will begin processing a medical waiver on your behalf." He does not have an LOA but was previously told that LOAs have their waiver process start automatically too.
I would say the difference with your child is that he/she had the LOA, got the waiver and that waiver carried through to the second application process. But according to USNA medical personnel, without the LOA your waiver won't be started until you're otherwise accepted - which I suppose could also technically be an LOA with the medical condition on it.