True about the swords SS, but not true about the armour.
The modern infantry lugs around with twice the weight of armour.
A knight who couldn't remount in battle lost his primary advantage so wouldn't have accepted the notion of calling for his winch crew during battle.
I'm guessing that the follow on from horses probably gives the best explanation for the side stand though.
Or maybe cos most people are right handed, they'd be right footed too so easier to stand on your left foot and do the intricate work with the right?
Good question though