I think warfighter crept into the lexicon for somewhat understandable reasons, likely because of the increasing frequency of joint operations (i.e., operations involving more than one branch of the military working together) after Vietnam, combined with the long-standing military tradition whereby members of any given branch take great offense if you refer to them using the wrong professional label (i.e., soldier, sailor, ~crayon-eater~ marine, airman, space cadet). That is, we can't just call all of them soldiers because only members of the Army are soldiers, so if for example you call a mixed group of marines and soldiers "soldiers", the marines will make their displeasure known to you, aggressively and in no uncertain terms.

When you're talking about DoD stuff all day long and frequently need to refer generically to the mixed personnel involved in a joint operation, warfighters beats saying Soldier-Sailor-Marine-Airman-Spacecase. All the other alternative phrases for the concept of "person employed by the military in one of the five combat arms branches" are variations on "member" and tend to sound clunky or be overly verbose, like "service member" or "member of the military." Try saying "service members" 50 times per day. Trust me, it gets old fast.

And frankly I don't see the problem with warfighter. Fighting wars is quite literally what they do, and pretending otherwise does a disservice to the truth and risks papering over the deadly seriousness of their work. Warfighter is also quite distinct from "warrior," which carries connotations of a specifically aggressive and barbaric flavor of professional violence purveyor. Like you say, it sounds like some atavistic hereditary soldier caste for whom violence is a sacred vocation joyfully undertaken rather than a solemn duty carried out only with great reluctance and forbearance.