> But it adds the whole second gas generator and turbine. No, it doesn't lower part complexity as a whole :) .
Because the fuel side and oxidizer side typically have a very different flow rates, many engines have separate pumps and turbines for them on separate shafts even if both sides use same fluids in the preburner. For example, see the SSME.
With FFSC, each of the turbines have no sealing requirements. This means, among other things, that bearings become easier and safer, as you can afford larger margins in your design. In many ways, FFSC allows the individual parts of the engine to be simpler.
The main cost of doing it over gas generators (other than oxidation-resistant superalloys...) is that all parts of the engine sort of circularly depend on each other -- when the Merlin engine was in very early development, they drove into the desert and ran the gas generator and fuel pump on it's own with basically no support infrastructure. Testing Raptor in a similar way is just not possible. It would have never gotten built without the NASA Stennis facility that can provide all the intermediate fluids at the pressures required.
I fully agree with the rest.