Or we could extract it and not burn it, making it "carbon negative".
Though your comment has a problem. It isn't carbon that is the problem, it is carbon in the air. If we did 100% CO2 extraction from coal fire plants they wouldn't be a real climate issue (overly simplified). Similarly we can't consider sewage as a carbon 0 cycle because eating turns a solid carbon source (food) into a gas. Carbon is fine, carbon in the air is not fine. Sewage is only neutral in the respect that we've already converted it to a gas (unless we accelerate this process, which is typically done) but doesn't account for the conversion process that happens.