In a hydrogen engine, you can do fuel rich staged combustion.
With kerosene, you must go oxidizer-rich. Nobody has mastered that except Russia. Nobody really knows how much it costs, or how they came up with it.
SpaceX uses the far simpler gas generator cycle.
In staged combustion, you pump some of the propellants to some absurd pressure like 300 bars, burn them, put them through a turbine that extracts some power that lowers pressure to 250 bars or so. Then the turbine exhaust is put into the main combustion chamber, together with the other propellants that were pumped up to 250 bars. All propellant contributes to thrust.
In a gas generator, you pump all propellants to perhaps 70 bars, divert some small part to a gas generator that burns them, the exhaust is put to a turbine that exhausts to ambient air. The turbine powers the pump. So that part of the propellant flow does not contribute directly to thrust.
If you go to higher pressures with a gas generator engine, you waste a larger portion of the propellant to run the pump and it's just not worth it. So gas generator engines are limited to much lower pressures than staged combustion engines.
Lower pressure means bigger engines are needed for the same thrust. Also the efficiency is worse. A large (tall) rocket can have problems with limited base area per mass, hence for example the low pressure engined Saturn V had an expanded base.
Space Shuttle's main engines ran with high pressure and could fit on the back of the relatively small reusable orbiter vehicle.