Field of view and aperture are different things. A 50mm lens has the same focal length and the same f/2 light-gathering ratio regardless of sensor size; APS-C shows a narrower view only because the smaller sensor crops the image circle.

The aperture diameter is tied to f-number and focal length, not sensor size: 50mm at f/2 implies an entrance pupil of about 25mm. That controls image brightness/exposure, not angle of view.

Using a full-frame lens on APS-C does not mean only the center of the front element is used. Light from all parts of the lens contributes to the image formed on the sensor. APS-C simply records a smaller central portion of the full image circle.

What makes many APS-C lenses smaller/cheaper is that they only need to project an image circle large enough to cover the smaller sensor. So the rear optical groups and overall lens design can often be smaller or less demanding. It is not necessarily the front element alone that is reduced; the difference is mainly in the size of the image circle the lens is designed to cover.