I've heard that juke boxes also keep a count of how many times each song is played, and this information also gets reported to ASCAP and BMI. (Maybe not all juke boxes, maybe just a statistically significant sample of them?)

That is true; however, even if no royalties were involved, the juke box operator needs that information to determine which songs to keep and which to replace, based on number of plays.

There was a similar thread a few months back complaining that royalties were driving a dance studio out of business. The amount they were required to pay averaged less than two bucks a day! Gimme a break! How much do they charge hourly, per student, for dancing lessons?