it is suggested that mounting a horse from the left was based on where swords were worn, etc.
This undoubtably is a factor but it misses a much more basic reason. Most people are right handed. If you are right handed, just try to get on a horse from the right side and see what happens. Right Handers typically do heavy work with the left hand/foot (even if the right arm is stronger) when they need to use the other hand for fine work (reaching, grasping). Watch a right handed high jumper, basketball player, etc and you will see he always pushes off from the left foot when he jumps. Try to jump on a horse from the right side by pushing off with your left foot and see what happens. You will look both like and at a horses ass.
“to answer the question, we need to delve into equine physiology and psychology” is a lot of horse manure. All you need to do is try mounting the right (off) side of a horse with something hanging from the left side of your belt, and you will see for yourself that you get stuck. I sometimes carry a radio, or a saw and scabbord on my belt (left side because I am right handed). If I make a mistake, and try to mount the off side here is what happens - my left leg swings behind to clear the horses croup, and over to the near side but the scabbord on my left hip stays hanging on the off side. I cannot simply move the scabbord to the near side of the horse, because both my hands are occupied. My right hand has the reigns, and my left hand is on the pommel, helping me lift myself up. It’s as simple as that. There’s nothing to do but go around and mount the near side. It has nothing to do with spooking the horse, or any nonsense like that. You just cannot mount because an object hanging from your waist or belt drops, and blocks your way.