There are actually a variety of leaf shutter designs, so a single answer might not be entirely appropriate... but it seems apparent that you are talking about the in-lens aperture diaphragm type.

There is an error in your understanding. The central light path does not expose the central portion of the sensor/negative... e.g. using a small f-stop does not result in a small circular image in the middle of the image area. Instead, the image circle remains the same size and is limited by the lens barrel.

The first thing to understand is that there is all of the light required to create a complete image at every point on the objective lens; i.e. the light from every point in a scene spreads out and hits every point on the lens.

And aperture adjustment is really just changing the exposure by the stacking of multiple images upon the image plane (fewer/more). This type of leaf shutter does not cause mechanical vignetting... the only mechanical vignetting will occur at max apertures where images taken from the edge of the objective element are included; and those images include the edge of the lens barrel (which causes the shading; and is cured by stopping down).

Some other types of leaf shutters can cause mechanical/optical vignetting if they are outside of the lens assembly (more like a focal plane shutter or apodization filter).

Some lens designs can suffer from natural vignetting due to the inverse square law (cosine fourth law) where the light from far off-center portions of the scene have to converge at the center of a small diameter lens and then spread back out widely over a short distance in order to cover the image area... Basically, wide angle lenses with short flange distances are most susceptible. But it is not caused by, nor cured by, stopping down an aperture restriction.