Incomplete combustion is a big component of emissions, and it's exactly what you're saying doesn't exist

Yes but since incomplete combustion is inverse correlated with fuel efficiency (unburned fuel is wasted fuel), it's not really a trade off. What is a trade off is NO emissions vs fuel efficiency. Burning your fuel oxygen rich will burn of more fuel, but also makes more NO (due to higher temperatures if I remember correctly).

Those eventually degrade to CO2 so the increased warming from them compared to co2 by mass is temporary, like with methane.