I think people are presuming that there wouldn't be public health engaged in the vaccine.
When I went to get tested for COVID, public health was doing everything from registry to testing.
Picture the vaccine:
It has to come in, it has to be distributed, it has to have people being advised that they can get it, it has to have controls at the vaccination site. Actually, the vaccination sites need to be setup. People who are working need support. At some point, someone needs to get the shot. Much of that other work can be done by the logistics / distribution of the armed forces. The actual shot is a very small part of the vaccination process.
There are also nurses in the military. There are paramedics in the miltary. There are doctors in the military --- including all the reserve units.
Might make sense for people to educate themselves on what our army resources look like, and what our reservists look like, as well as what is required, prior to being critical of the concept.