It's not that. It's that, unlike you, most of us do not have a worldview wherein each individual is just an individual.
According to you, if I believe in it, I should go fight for it, and furthermore, you find it unbelievable that anybody even believes in it, because how could this ever affect our lives?
This is human suffering on a totally unprecedented scale. This is full-blown genocide, targeting both ethnic and religious "others", to a degree that's only been attempted maybe five times in human history.
Most of us believe in something called a humanitarian obligation. There is a line, everybody has to draw it for themselves, but there's a line, and once it's been crossed, the rest of the planet has a moral obligation to end the violence if they're able.
We're taking it in pieces, which is responsible, because we've just been mired down in a war in the Levant for ten years. So we begin with airstrikes, whatever. I don't think that'll be very effective, either, but at least we try it.
So. Why should we care if millions of civilians are being shot in the back of the head because they aren't Sunni Arabs?
Because fuck you, that's why. Because FDR said when your neighbor's house is on fire you don't haggle over the hose. Because we've seen what can happen when you let this kind of organized killing go unchecked. Because, for the first time maybe ever, the people who are doing it aren't a world power; we aren't powerless to help.
At this point I think most rational Americans would agree with you that most deployments of the past 50 years have been to nobody's benefit but the elite. This is different, though. Much different. And, even if it weren't, the elite has been deploying troops to nowhere to fight for nothing since WW2 ended.
So I'll say probably the rudest thing I've ever said in my life: if you joined the military despite being philosophically opposed to ever being deployed, you made a bad decision, and that's okay. But if you still think your enlisting was a good decision, if only that pesky populace would stop permitting the President to deploy you against your own judgment, well... you're a fucking moron. I didn't put you in the military. You put you in the military.
99% of troop deployments make me viscerally angry, I've been out in the streets holding big signs begging leaders who don't listen to bring you home, but because one time I think we should get in the fight, because I'm not myself going over there, that's it, whatever else I believe is irrelevant, I'm a coward?
Fuck you. Super glad I advocated for you. Entitled little sack.