The Final Chapter - I was back at the part's store, talking to a different associate about the column switch. He said the "click" should come from the switch itself and I can swap it for another one. Go home, take off switch, come back later. Turns out my first switch worked fine, once you remove the
packing brace There weren't any instructions with the switch, and apparently the associate who sold me the part in the first place didn't know about removing the brace either.
Soooooo, after removing the original bad switch, installing new switch #1 (with brace), buying foot switch, removing new switch #1, returning new switch #1 and foot switch, installing new switch #2 without brace - TA DA!