Not really, no. For optimum propeller performance you would need to maintain an angle of attack on the propeller blades close to the optimum lift over drag ratio (around 2/4 degrees AOA, similar to an aircraft wing). Flying at fine pitch (high RPM) at higher speed or vice versa will result in lower or higher than optimum angle of attack, reducing propeller performance. Done correctly the balance between MAP, RPM and airspeed will result in the propeller getting roughly the same “bite” of air during climb (low speed) as during cruise (high speed).
On a fixed pitch propeller, depending on which phase of flight the propeller is optimised for (climb or cruise) the “bite” of air will only be right for that particular phase of flight. Most propellers are optimised for cruise performance as the aircraft spends the majority of the time there, in such case the “bite” of air will actually be bigger during climb as compared to cruise (exactly opposite to your theory).
The whole comparison between a constant speed propeller and the gears of a car is a gross oversimplification. The bite of air theory as used in one of the articles is also not entirely correct. The “bite of air” the propeller takes is basically the angle of attack, not the blade or pitch angle (blade or pitch angle means the same by the way, contrary to what is written in one of the articles). Its difficult to explain but you need to visualise the path the propeller blade is moving to understand what “bite” of air the propeller is really taking, a higher blade angle does not necessarily mean a bigger “bite of air” if the aircraft is flying at a higher speed.
Not completely on topic but it might clarify some propeller principles: