" I cannot understand how the "fuel pump" actually pumps gas."
The pump is activated by the engine crankcase pressure.
As the engine rotates there is a pulse made.
This pulse pumps the fuel pump, which intern moves the gas from inlet to outlet of the fuel pump.
Sometimes a new pump is dry.
Try putting some gas into the outlet side hose of the new pump to help wet the valves inside it and see what happens.
Leave the outlet hose off the carb to watch for a pumping action as you crank.
Also make sure the crankcase pulse line is in good shape, no cracks, ect.