To spare me going through all 100+ sites to verify my hunch, can someone demonstrate either that it is the rarest badge (possibly tied at 0 with other rare badges) on every site in the network, or provide me a counter example?
It is not the rarest badge on every site.
Over on Arqade, it has been awarded once, whereas Tenacious and Unsung Hero have never been awarded. There are similar situations with those badges, or Reversal, on other sites as well. Electrical Engineering was mentioned in the comments, and I also spotted it on ELU.