The purpose of leg drive in the bench press (force body diagram) A common explanation I've heard for the purpose of the leg drive in the bench press is that the lateral forces generated by the quads off the floor are "converted" into a downward force into the bench, at which point, because your shoulders stay still on the bench, there is an upward normal force that is transmitted through your arms, into the barbell. Because of this, your arms uses less force (as your legs are now doing some of the work).
My physics knowledge is rudimentary, and so I was hoping anyone could shed light on this phenomenon. And if the explanation is wrong, if anyone could clarify the nomenclature for me, and if anyone could help me out with the force diagram.
I found this (incomplete) force diagram on the web that is hopefully representative of my understanding of how a lateral force from the quads helps with the press. Many thanks for the help!

 A: Unfortunately, its rare that we can answer questions about the body with rudimentary physics.  We can draw free-body diagrams of humans, but it just doesn't provide the insight we hope for.  This is because the human body is an incredibly complicated system of muscles and feedback loops.  We are almost never operating at 100% of our muscular capacity -- we typically have reserves to deal with stabilizing our body.
And, in this case, it appears that is why there is the leg drive.

Leg drive is when you drive your feet into the floor as you push the barbell off the chest. Leg drive in the bench press will help you maintain your upper back position and increase the stability and stiffness of the torso, which allows for greater levels of strength.

You can read the article, which goes into far more detail than I will here.  But the gist is that the leg drive helps properly position the head and shoulders so that you can get the maximum strength out of your muscles.  It also helps with arching the back, and I won't even touch that one -- the physics of the spine is astonishingly beautiful, but painfully complex.  If the experts say you want an arch there, I really won't question them.
It doesn't look to be actually "lifting" the bar with the force of the legs.  It looks to be all about positioning muscles and ensuring they have the support needed to be stable.
A: This answer is more me having fun playing with the toy model than trying to answer the question directly, which I think Cort Ammon has already answered.
Much of the effectiveness of doing this kind of technical lift is surely in greater body stability, as Cort Ammon explained. But I think a toy model can explain why the weightlifter feels the sensation he feels, and unless I've made a reasoning error, it also highlights another reason why the technique is useful that is missed by the biomechanics approach.

Replace the human with a rigid bent rod on a pivot as shown. The pivot is the center of mass of the human-barbell system. The person's "head" is on the right, their hips are on the left, and their feet are at the bottom.
Constrain the pivot such that it cannot move left or right (it is held in place by static friction) and such that it cannot move upward unless the net force on it exceeds the combined weight of the human-barbell system.
Place an immovable barrier (the bench) under the horizontal portion of the bent rod.
Apply a force $F$ at the the "feet" end.
Note that this creates an equal and opposite force $-F$ at the "hips" end.
Note that this creates a torque $-F sin(\theta) A$ about the pivot.
Since the bench prevents the rod from rotating, there is an equal and opposite torque from the other side.
Therefore there is a force $F sin(\theta) A/B$ compressing the "shoulder" end into the bench.
This is the force felt by the weightlifter, who feels "like I am shoving my upper back 'down' into the bench when I use leg drive".
There is a direct consequence of this force that makes it easier to lift the bar: it moves what one might call the "effective center of gravity" of the system closer to the axis of the lift.
(I think it's more correct to say that it sets up a torque such that when the weightlifter and the barbell separate, the lateral shift in the position of the center of mass creates a proportionately smaller felt shift in the direction of the gravitational force vector... but that's much harder to visualize than making up something called the "effective center of gravity" and letting it move.)
We can diagram this out using the same toy model by getting rid of all the lateral forces (we know they are neutralized by static friction) and supposing that the body is made of three point masses located at the feet, hips, and head, while the barbell is another point mass.

Since the feet and shoulders are now "heavier" and the hips are now "lighter", this shifts the "effective center of gravity" closer to the head.
And, as with any heavy lift, the closer to the axis of the lift the weightlifter can get the center of gravity, the easier and safer the lift becomes.
