Q: What does this pivot pin do?

It's the pin/shaft that the pedal arm pivots on inside the pedal support -- the wear part that rides in the pedal bushings. Forum discussions on pedal pivot shaft replacement describe replacing it, along with fresh bushings, whenever the original pin is worn, bent, or has to be removed.

Q: How is the old pin removed?

Forum builders describe grinding the swaged end of the old pin down close to the pedal, supporting the pedal arm in a vise, and driving the pin out with a punch (or an air chisel) rather than simply unbolting it -- expect to cut and drive the old one out.