Lots of presumably learned research and obscure historical fact in that article. But it continually states that eeny-meeny is a 'counting system' eg. "used by shepherds to count their sheep" etc.
Eeeny-meeny is not a counting system. Why would people use a nonsense rhyme to count things? It makes no sense (literally!). Eeny-meeny is used as a 'random selection system' when choosing arbitrarily from a series of options.
Odd to get that basic fact wrong in such an in-depth article.