Why do we care? Because this is just the latest installment of our perennial war. Most of the mob are now in the "Je suis Ukraine" phase of their existence, but as you well pointed out, they would not care if it wasn't for the media instructing them to.
But I also think we care for a very selfish reason. We are tired. Talking to my peers (so early-mid 20s), I get an overwhelming sense of "this can't be the end" attitude, and I'm sure it's replicated, perhaps to a lesser extent in older generations, as well. We're coming off of two very stressful years, regardless if you saw through the whole BS or not, and now a war? I think the reason most of us care is that we just want our lives to go back to normal already. Of course we don't actually care. Sure, what's happening is a tragedy for the Ukrainian individuals caught up in this, but I guarantee you that 99% of European/American people watching this (at least) would gladly trade in many more dead in Ukraine, just as long as they can go back to their usual life.
Who's going to fight it? Gen-Z? I think we can all admit that this seems... unlikely... and borderline laughable.
I keep saying the same. Young men today continue being children, essentially, well into their 30s. It would be a depressing thought to think they're the ones who will defend our countries, should push come to shove. Not that young women are better.
Are we in the "Hard times create strong men" phase?
Yes, at least I think so. Then again, we could be in the "Lies and deception create obedient men" phase.