> 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.

By "many" you perhaps mean "many American". In Russian engines it's mostly a single shaft.

> 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...

Yes, rocket engine is usually a complex dynamic system, with deep feedback loops. You can still get gas generator and turbines with pumps tested separately, but you need to measure dynamic properties - like resonance frequencies - in order to have better chances of the good work when the system is integrated.