There was No Confusion, the students were told they were admitted on Official USGA letterhead, which is pretty clear. The fact they were just then denied without even an apology or other option is very unprofessional. The school needs to open up an additional week and give those students a shot or at least add them to the top of the Waiting list.
The Service members in my family don't believe this and are upset.
Relax. First off, there was an apology, you quoted it in your post. Second, there are other options for applicants to decide if USCGA is the right fit for them, which is the purpose of the AIM program, including but not limited to the Cadet for a Day program. USCGA is run by humans and humans are bound to make mistakes.
I can relate to the frustration and disappointment applicants are feeling who were denied, but at the end of the day AIM is just a week long program. In no way is it going to decide their success in applying to USCGA the following year. This isn't going to be the last time these students are told something only to have it be retracted later throughout their life, especially if they are seeking a career in the military. Ultimately, the needs of the service are going to take priority over their feelings and they had best get used to that now. Build some resilience and focus on the main goal, which should be admission to the Academy, not a week long program in the Summer.