> ...it also lowers part complexity, because it eliminates interpropellant seals.
But it adds the whole second gas generator and turbine. No, it doesn't lower part complexity as a whole :) .
> It improves achievable chamber pressure
Yes, I'd even argue that's the penultimate goal of full-flow scheme.
1) For a rocket engine you usually (almost always) want as high Isp (which is approximately the speed of gasses flowing from the rocket engine) as possible.
2) To get that Isp, you need as high pressure in the chamber as possible.
3) To get that high pressure, you want to supply your pumps with as much power (in Watts) as possible from the given overall fuel flow (in kilograms per second) - while not melting turbines blades (so fuel:oxidizer ratio is limited).
4) To get most power to the turbines blades at fixed (maximum) gas temperature and fixed overall fuel flow, you want to use all of that flow (classical staged combustion uses only one component, which is essentially not using the whole fuel flow) and adjust ratios fuel:oxidizer at both gas generators so that total power would be maximum. You want maximum flow both because that increases turbine efficiency and because that actually provides more power to the turbine.