I am pretty sure AMHA rules dictate the side the handler walks on....it does dictate the direction in which you walk.
You have to understand where we are coming from, the WHOLE American way of showing is crazy to us as we place so much emphasis on way of going...which does NOT mean "let's see how fast or flashy our horse can run" (any breed, BTW)
but "let's see how good the conformation on our horse is when it is moving"...so running up and down a rail would not do it for us.
In European showing the Judge gets a chance to view the horse walking and trotting away, and can thus judge the hock action- which it is not possible to do when a horse is standing still, and the conformation of the horse- it is almost impossible to cover conformational flaws on a moving animal.
At no time except maybe up the back of the line up, is a horse required to extended, these are breeding classes, not speed classes.
I do hesitate to criticise other countries methods of showing, as everyone does things a little differently, but the American way seems to be based on glamour, speed and hype, and speaking personally, I loathe these things.
And I showed Arabs for years, I had no problems with ring manners whatsoever, the French Arab show does have people hurtling along in front of the horse (fit young men in trainers, not demented dwarfs!!!) I just see no advantage to it.
I find the American shows really interesting to watch, but I am constantly glad I do not have to do it that way.
As Oubergia said, just be walking clockwise you could make the whole thing so much easier all round, and the Judge would be able to see everything so much better.
Which, of course, leads us to the conclusion that, as with the measuring, maybe this is why it is not done???