Fossil fuel is carbon that is stuck deep in the ground, we pull out it out and burn it in airplanes, that adds carbon to the atmosphere because otherwise it will have just stayed in the ground.
When plants or trees grow, they suck carbon out of the atmosphere to create their structure. All of the carbon in a plant or tree comes from the air. You harvest* those for food (or SAF), then you burn it in your stomach or jet engine and release carbon dioxide, but it is exactly the same amount of carbon that was in the plants, which was all taken from the atmosphere, using solar energy. So it’s a closed loop.
The harvest and processing is where things go wrong as Jim points out above. All the energy required for harvest and processing may require carbon, which means it has a carbon footprint unless you are using nuclear or solar or whatever for that part.