Grandma says no woman from her childhood ever rode a horse any other way than how the men did, which was straddling a saddle or bareback.

I reckon if a woman were to need to mount on her own without a step up aid and wearing a tight unsplit skirt, she'd be compelled to hike that skirt up high enough to get her legs apart so that she could mount the saddle using the left stirrup and throwing her leg over to straddle it temporarily, then curling her leg back up and under the horn proper while adjusting her sitting posture accordingly. When I was a kid we tried to see what riding side saddle would be like using Grandad's high horned spanish saddle (as he referred to it, he liked it for roping as it gave his arthritic hands a little more leeway). Basically you had to mount the saddle same as for regular riding, then swing your right leg back as I described. You could do the same thing bareback long as you could grab a handful of horse's mane and tuck your leg beind that, but then you had to hold onto that mane to keep your leg in place, and you dang sure didn't trot or canter in that position or you'd giggle yourself outta your seat.

We concluded sidesaddle was for women who were more concerned with how they looked than whether they could actually accomplish anything on the back of a horse. Perhaps there is some purpose in it, but no girl I ever met that could ride would ever likely consider it as anything but a novelty.