Did anybody else figure this out?

Yeah, it is ironic in a way, they each try to kill the other, unaware that the one they're urging to kill is necessary for their own existence. It's an interesting story theme, one of the many reasons I like the Termie films!

Oh wow, Ты think too much >.> ANYWAY

I think the best & simplest answer there is, is that John is a hero. He does care for his own existence, for certain, & I'm sure (in that little world & story) that he's had plenty of time to think & already know about this. He tries to destroy Skynet to save the world, even at the cost of his own life. That's it.

It's sort of a paradox that would happen with any form of time travel. Hawking сказал(-а) that that the time-stream would do something like this to fix itself. So, it's pretty much nature in there too.

If there was no Skynet, then there would be no John Conner. It's because they would eventually make the time travel possible that John is even around. I know, it's hard to think about. But that's time.

Ah the paradox that every time traveler faces, but alas without skynet, I, the world's first robot butler would have never come into existence and neither would my website!
link

honestly, i used to think about this too
truth is; one can't exist without the other.
John can't exist without Skynet, because if it wasn't for the terminator, he wouldn't have born. Skynet can't exist without John, cause if john wasn't born, skynet wouldn't have sent the Терминатор which caused john to exist in the first place.

so there Ты go..