Neither can live while the other survives, though. If this theory was correct, it'd be closer to say Neither can live while the other dies.

On the other hand, since we haven't had any Voldemort point of views, we *don't know* that Voldemort knows this or not, so only your point 1 remains. I'll note that Draco was sent to kill Dumbledore, not Harry, and Snape apparently wasn't instructed to kill Harry off either, at least not as a priority.