It never concerned me to get on from the right side. I get on from a mounting block for ease on his back.
THEN I came off him and badly hurt my hip. When I was sound enough to get back on, I wasn't able to swing my right leg over him (even when mounting from the bed of my truck). Obviously I wasn't completely healed, but was unable to stay away from riding.
I can and do now get on from both sides.
The reason for mounting from the left side, goes back to the army days. It was safer having everyone mount from the same side. I read this somewheresss.
I think it's good for soundness issues, for their back, and to help if you aren't sound or in a situation where you can't get on from the left.
...This doesn't mean I'm capable of mounting from the ground from the right side. My brain still isn't capable of mastering it.