Transmission

It crossed my mind when I saw that.....

Yes, it can reduce the effort to pump the pump arm per-stroke, and do so dramatically, BUT....

It also greatly increases the parts count, and complexity.
It adds losses. Each point at which metal contacts metal, ( or anything else for that matter ) produces friction. Even the internal molecular shear of oil or grease is a friction loss, small as it is.

The question(s) become how many strokes, at any effort, is too many ?

A one-stroke 2000:1 compression ratio pump, if possible, that takes 200 strokes at the arm, is still 200 strokes at the arm.
Without the additional losses of gear reduction, it might be only 150 strokes, but at slightly higher physical effort.

Still, it's ideas like this, no matter how "out there" that produce innovation.
Can't even begin to tell you how many bad ideas I've had that eventually led to something quite workable.