I asked this question of a Budweiser wagon driver once, during a tour of the brewery and stables. He said it was in case the driver passed out - the donkey would still be able to drive them all home.
Just to sort out some of the nonsense regarding counting rhymes:
The idea that children keep ancient memories alive in their games is neat but it isn’t historically accurate. Anyone inclined to mention “Ring around the Rosie” can find a nice debunking on Snopes.
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Yan-tan-tethera an other “sheep-counting doggerel” are clearly related to Welsh and the other British Celtic languages, though they aren’t necessarily ancient. However they got started, only about one number in five is actually a recognizable Celtic numeral. The others are nonsense syllables created by generations of people who had no Welsh (a lot of the pairs of numerals rhyme, and some are out of place; there’s a distinctive poetic rhythm). Likewise, “hickory dickory dock” may be “8 9 10,” but only “dock” looks like a memory of a real number.
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Eenie-meenie-miney-moe cannot be Celtic. The series would probably be “1 2 3 4,” but of all of those only “eenie” looks like a number. “Een” is a decent approximation of Welsh “un” (1), but -ie is an English suffix and the other three don’t map onto anything at all. In its current form it is most certainly not a survival of anything, but it’s not at all unreasonable to see it as a series of numbers used by children for counting out.
How do you make the change? Will you have a transition where you call both terms so the gee/haw dogs and the left/right can all understand? What happens when you want to sell a left/right dog but everyone else uses gee/haw dogs?
I like the hypothesis that Gee goes up and Haw goes down so they are easily distinguishable. In the same way, herding dog directional commands are Away To Me (counterclockwise) and Come Bye (clockwise). The whistle versions for these are high/low and low/high. These are used all over the anglophone world.
I’d like to point out that German commands for protection dogs came into use in the US because so many trained protection dogs are imported from Germany (and now, the Czech Republic and Belgium). It’s a thriving industry there. You don’t want to change their command words. Scotland exports a lot of trained sheep-trial Border Collies to the US, and it isn’t uncommon to hear some Texan sheepdog trialer crying Lie Doon! to their imported dog.
My grandparents (Wisconsin farmers) called pigs with “Soooooey!” and cows with “Come Boss!” I think these are also widespread over the anglophone world. The inherent conservatism (meaning the conservation of habits and traditions) of people who are close to the land means that these things don’t change unless the chain of instruction is entirely broken.