I just found this incredible video KSP Doesn't Teach: Rocket Engine Plumbing and ran through it a few times. It's very helpful in the way it outlines the various issues first, and then designs the designs that address them.
For a Full FLow Staged Combustion Cycle engine there are two combustion chambers driving two turbines; one to drive the fuel pump, and the other to drive the oxidizer pump. This "two shaft" solution solves problems that can be caused by leaky seals as explained in the video.
If I understand correctly, these combustion chambers are run either fuel-rich or oxidizer-rich so that they can deliver enough power without running so hot that the turbine could not handle it. By running on either side of the optimal mixture, the unused gas still expands and produces power by absorbing some of the heat generated by the reacted gases. And since the system is closed, it gets used later anyway.
But I am still having trouble understanding this diagram. If oxidizer and fuel are both cryogenic, why is it only the fuel that gets pre-heated? And are there additional propellant feeds from the pumps to the main combustion chamber, or does ALL of the propellant enter via the turbines in the form of incompletely combusted exhaust from the two smaller combustion chambers?