"Awkward and bureacratic" is literally the point of naming conventions commonly adopted by democracies. Titles like "president" or "prime minister", departments like "Department of Defense", referring to government employees as "civil servants", etc. are all intentional measures meant to strip away the prestige and egotism associated with positions of authority in an effort to avoid it going to people's heads, and to remind them that they are meant to serve the good of the public that pays for their existence rather than ruling over them.
"Service member" is awkward, because it has too many syllables. People won't use it when shorter alternatives are available. And it's bureaucratic because it's unspecific. It doesn't tell anything the service those people are members of, and it doesn't tell what kind of work they do.