Even if the vaccine _only_ protected the recipient (from severe illness and death), it would still be in the Navy's interest (and authority) to require sailors

Even if the vaccine _only_ protected the recipient (from severe illness and death), it would still be in the Navy's interest (and authority) to require sailors to receive it before deployments. This is no different than all the other preventative care non-civilians submit to in order to avoid having health problems jeopardize the success a mission later on.