Congratulations, you just invented Merged Mining: you can mine Bitcoin, and your Negative Coin, while working on the same block headers. Some of your hashes will be accepted only in Bitcoin, and others only in your Negative Coin. But because there is no overlapping, then you will have no block, which would be valid in both networks at the same time.
why don’t we consider passing them through a simple NOT gate?
Because then, your blocks will be valid only in your network, or in Bitcoin, but never in both. Usually, people want to mine blocks, which could be valid in both networks, because then, they can mine their altcoins, and see the progress, while also having some low-but-never-zero chances of hitting a valid Bitcoin mainnet block, and getting 3.125 BTC (at the time of writing) plus fees. By making it strictly separated, you just make it harder to make swaps between your altcoin, and Bitcoin.