How does the throttle position affect the speed of airflow to the engine?
That's because it's not the restriction created by the throttle plate itself, that you see as manifold pressure (or vacuum), that is creating the suction that pulls fuel from the carburetor float bowl.  It's the restriction of the venturi, where the main discharge nozzle is, which is upstream of the throttle plate.  Since it's in in the upstream flow, the suction created at the venturi, being a fixed cross section, is a function of air velocity entering the carburetor, which is a function of throttle opening and engine speed. 

In other words, the fuel metering system at the venturi doesn't care about the pressure differential created by the throttle plate farther down, it only cares about the velocity of the total mass flow that engine is able to pull past the venturi upstream of the throttle.  

Open the throttle, the velocity of the air going through the venturi goes up, the suction generated by the venturi increases, and more fuel is pulled into the discharge nozzle.  And vice versa.