On the Lyc (Bendix) system, the pump is merrily pumping away, potentially supplying more fuel than the engine needs. The FCU simply limits that output via a diaphragm which responds to pressure drop across a venturi, to arrive at the desired flow. The pump can go as hard as it likes, but the FCU controls flow. That's why the flow doesn't change when you switch on the boost pump and raise the input pressure.
Sorry, but you lost me ages ago when you claimed that TCM and Bendix systems work the same.
Correct for the Lycoming! Well said. Also very common on the wonderful old big radials of yesteryear. Run the engine RPM up and down, and the fuel will change via the Bendix Pressure Carburetor and airflow.
Most CMI (Continental) engines have a MUCH simpler system where the engine-driven fuel pump is driven by the engine (direct gearing), and the output is variable with engine RPM. Run the engine RPM up and down, and the fuel flow will also change, directly with the RPM.
Both systems have their advantages and disadvantages. For the most part, they are transparent to the pilot.