I used to do that at a jewelry store and later at the jewelry counter at a big box store. There are sets of micro screwdrivers that have a screwdriver small enough to push out the pin. I had never seen the tool for it before your post, interesting to know that there is one.
I didn't realize this tool exist until recently. Used to think it would be difficult and delicate to resize watch band, but it turned out to be easy.