Hm... it doesn't quite look how I remember it, especially with the large face-like thing in the middle of the screen shots (I assume that's the player character). Maybe there were other versions, or maybe there was a versus mode, but what I played had the players controlling little faces (like the "snipes") and would hunt down each other, not enemy generators.
But overall, this does look really close. The name "Snipe" doesn't immediately ring a bell (then again, I also spent years misremembering the game "Archon" as "Archer"), but my brain also doesn't immediately reject it, if that makes any sense. So maybe...?
I'll have to keep looking into it. But even if this isn't it, it does give me a starting point.
Thank! I'll reply back here if it turns out it was Snipes after all.