"The first known examples of a team game involving a ball, which was made out of a rock, occurred in old Mesoamerican cultures for over 3,000 years ago. It was by the Aztecs called Tchatali, although various versions of the game were spread over large regions. In some ritual occasions, the ball would symbolize the sun and the captain of the losing team would be sacrificed to the gods." https://www.footballhistory.org/

Seems like one ball/rock was enough for the team captains.

With every fake handoff, American football takes on a second ball momentarily. What makes it interesting is when it isn't actually there.

But to your point, there are the Globetrotters who use multiple balls to make it interesting... though this may be outside the definition of sport.

I have a feeling the different sports corporations/leagues have done the research on this. I can't find any offhand, but maybe someone else has more luck?

Maybe it's the beer sponsors that want the game to be as easy to follow as possible, enabling a higher alcohol intake? ;-)

If so, then wouldn’t having three balls in play be even more exciting?

Couldn’t you just watch two games simultaneously for the same effect?

It feels like this would make games harder to referee.