CupofJoe
Myth Weaver
I went to check out a riding school when I had the urge to learn to ride. In the English/hunting form the owner of the school explained that their beginner/training horses are used to people [usually children] getting on them in whatever manner worked... as long as the horse was okay [and they seemed to be very content and chilled horses].... I'm not sure about the English discipline's training practices; I mostly rode Western.
I got the impression that for the occasional/weekend rider getting on from one side or the other was more to do with the muscles in the leg being strong enough, the horses in the stables were pretty tolerant.
The more advanced horses [where you learn to canter, gallop etc.] were more traditionally trained, to accept mounting from the left. I think a few would tolerate it from either side but not many.