Dismal with science, I said!
Now, where is a competent scientific/technical mind? Paging GJ engineers and physicists!
Yeah, when it comes to mechanical advantages such as leverage, pulleys, gear reduction, there is always a trade-off, no free ride, i.e. you don't get the mechanical advantage for free.
For example, in the case of leverage, and using a pry bar as an example, less force is required to lift the object you are prying, but you have to move a greater distance; so the total work that you have to do is the same (a little more actually, due to inherent losses such as from friction). You might move the big rock you are prying on an inch, but have to move your end of the pry bar a foot.
The same applies to gear reduction. Your engine has to spin at higher RPMs to reach a given wheel speed. Pulleys that are arranged in a mechanical advantage pattern are the same way, i.e., you have to pull your rope a much greater distance than the load moves.
Keeping this in mind, you can test to see if a longer shank can possibly multiply torque on a screwdriver. Take two screwdrivers that have the same handle diameter, but different length shanks. Screw in or unscrew the same screw with both of them, and count the number of turns. You will find that the number of turns are the same regardless of the length of the shank (or more specifically, a given point on the outside of the handle will travel the same distance regardless of shank length), which means that it is impossible for a longer shank to provide any mechanical advantage, because if it did, that mechanical advantage would be free, and the universe would self-destruct or something (lol).
On the other hand, with a larger diameter handle (which
does multiply torque on a screwdriver), the outside of the handle travels a greater distance to turn a screw a given distance compared to a smaller diameter handle, thus, you're paying for your ride.