The response from USPS is what I expected. It's unlikely this was in the system for 100 years, but someone put it in the mail recently and the USPS decided to deliver it.
Meanwhile I have significant less mail, and on mail that had a stamp I see it took them a week to deliver it (this is local mail, not international). Never seen I such issues with mail as long as I lived here.
It could have slipped behind a counter at the original post office and been uncovered when that office was decommissioned 35 years later. Then it would have made its way to a newer sorting facility where either it could have been misplaced again or, more likely, been stuffed in a cabinet somewhere (as I doubt USPS was as adamant about delivering lost mail in the 1950's). That facility could easily be in operation until today - or decommissioned very recently.
No idea if anything like this actually occurred. But at 400M+ letters / day, USPS has the scale for unusual circumstances like this to be somewhat regular occurrences.
In the UK it was up to 12 times / day in cities!
Also, cf. "Going Postal" by Terry Pratchett.