Mr. Stewart's response elaborated on what I didn't.
My experience has been that even an long-time mechanic can have the occasional bad rivet-push. The advantage of the experience is recognizing this and addressing it.
From observation, most 5 and 6 speed chains (pin width greater than 7.4 mm) don't use peened pins. It's also very hard to source quick links in sizes wider than 7.3 and smaller than standard singlespeed. So for those chains there's no choice but to reuse the pin and link. Also, peening on 7.3 chains depends on brand and model. You can pop and push Sedisports to your heart's content, but might not work as well on a Uniglide. Below 7.1 mm, it seems they're all peened.
I've seen chains at the co-op that had 3-4 quick links on them. Probably wanted to be sure there wouldn't be problems.