That's wrong. They aren't independent. It's a closed loop system under most conditions. If the O2 sensor isn't providing feedback, the computer has to make guesses based on the rest of the sensors, but the O2 sensor provides actual, real-time results of how close the fuel mixture actually is. If the system is not sure, it will run on the rich side to better avoid potential damage from an overly lean condition. Potential valve damage, detonation, etc,
The fuel economy can vary significantly - as much as 25%, and that particular number is from personal experience.
On a Mass Air system car I owned (vs Speed Density, which by nature runs by somewhat different logic but still needs the O2 sensor) check engine light went on, car still seemed to be behaving exactly as it always did, but I wound up getting 19 MPG per tank vs my usual 24.
That $18 I spent for a replacement unit was more than worth it.