"For instance, in Final Fantasy VI, an entire kingdom is killed. AN ENTIRE KINGDOM. Pshh, we don't care about them. We only care about party members!"
Well, uh, yeah. Now this may sound insensitive, but I think you would be more personally affected if someone close to you that you know died than if a thousand people somewhere far away you've never met die.
That said, I think it's because FF7 is such a popular game that it overshadows other FFs. I've never played FF4, so I don't know about those kids dying, but FF4 is nowhere near as popular as FF7, which is why I think Aerith dying gets more attention.
Also, Aerith was very easy to like. I think that was part of it, it's been ages since I played FF7, so I might remember it wrong, but wasn't Aerith pretty much pure and innocent all-together? Seeing her die was a pretty big blow because of that.
And for my personal reaction, I played FF7 when I was like 6-7 years old and had a crush on Aerith, so naturally I was pouty as hell when she died and didn't want to play anymore.
