The reason so much is done on the horse's left side is because he processes things whereas the off side is his reactive side. This also fit in with riders carrying swords which hung on their left side, as it was easier to mount. To reduce how reactive a horse can be it's best to train both sides, not only equally well but often more so on the off side. This also builds muscle symmetry.
Aw man, that "sword" thing is fantasy land hogwash.
The vast majority of people are right handed. With right handed people, the left side is the strength side, thus the left foot is the "base". When you step over something you plant your left foot and step over with the right. When you jump, you plant and jump off your left foot. So, saddle or no, sword or no, the natural tendency is obviously to mount or jump on from the left side. To do it from the right side would be an unnatural movement. I don't know why people have to romanticize everything...
Faceman, while I don't think the reasoning behind the "sword thing is entirely hogwash or out of Fantasy Land" I do think you have "hit" on a very logical line of reasoning for mounting on the left side. Case in point, several years ago while out riding I had to mount and dismount several times always from the "near/leftside" of my horse. I decided at one point to remount from the "off/rightside". You know what!!! I could not do it, but I was 40+ years old at the time. I had far less strength trying to pull myself back into the saddle from the "offside". Then I further experimented with your theory on a right-handed person always taking a first step with the left foot. I actually got up from my computer chair and without giving any thought to it my first step was with my left foot. Then I attempted and had to concentrate on stepping out with my right foot first. It was not a natural feeling and it felt unnatural as all get-out.
As someone who used to regularly mount a horse with a sword (wooden practice) it really is easier to get on from the left side. Unless you are one of those bizarre left handed folks who wear the sword on the right side and they seem to mount on the right side of the horse.
Of course I used to be "one of those weird folk who play dress up medieval and bash at things with swords/lances/spears". Now that I just do "regular" riding I mount from both sides, whichever I am closest to when I get ready to get on.