From the online OED (yes, I can access it again!), the first citation is from 1602, meaning a worthless person. Then in the 1700s, all the way to the 2000s, it describes "any of various games which feature the rolling of a ball or other object" or a game where people (especially children) roll down a hill (1821 to 2000s). The first OED citation for "a podgy, plump, or stout person, esp. a child" was from the European Magazine in 1808, with this quote, "The eye has its fire and consequence correspondent with the fierté belonging to a man of rank; the good-humored rolly-polly which gave the appearance of twinkling stars to those organs, is no more." In other citations "roly poly" just has one L.
Proof, that BINGO story was on NPR's "Wait, Wait...Don't Tell Me!" show today. The game contestant didn't pick that story so they lost. Had I been playing, I would have won! [I think winning means they put the message on your answering machine. A million dollars would be much better!]
What would have happened if he'd walked in backwards and excitedly shouted, "OGNIB?" In the emotionally charged atmosphere of a bingo game, I bet ANY excitedly shouted word would have had the same effect.
It is no measure of health to be well adjusted to a profoundly sick society. -J. Krishnamurti