This Aerospace Corporation article explains it beautifully.
First, all of the propellants are burned in the preburners, thus
providing more mass flow for turbine drive power than the
conventional staged combustion cycle. This additional power can
be used to increase the chamber pressure and produce a
smaller engine; alternatively, the preburner temperature can be
reduced to provide the same power at lower temperatures. The
lower turbine temperatures translate into longer turbine blade
life—often the limiting factor on reusable engine life.
The second advantage is that the use of oxidizer-rich gas in the
oxidizer turbine and fuel-rich gas in the fuel turbine eliminates
the need for a complex propellant seal for the pumps. There is
little risk with leaking liquid fuel into a fuel-rich gas or liquid
oxygen into an oxidizer-rich gas. In contrast, the fuel-rich staged
combustion cycle must use sophisticated purges and multiple seals
in the oxidizer pump to prevent any liquid oxygen from leaking into
the hot fuel-rich gas. A similar situation must be avoided in
the oxidizer-rich cycle on the fuel pump side. The elimination of this
failure mode increases system reliability
Your concerns with "feed[ing] the full volume of fuel and oxidizer through the plumbing of the preburner and pump turbines" are not well founded. The "full volume" of propellants is going to be fed through the engine plumbing somewhere. You also state that "Also plumbing simply has to be larger and heavier"; this is not necessarily so; the general decrease in engine size and elimination of seal systems and helium tankage to pressurize them may reduce the overall weight. Also, this paper compares a Raptor-like engine with a single-preburner fuel-rich staged combustion engine; the pressures in the two engines do not differ greatly; it is not clear that one or the other requires heavier plumbing.