I own a side-saddle. I guess mine is kinda an antique, because we have had it forever. It probably dates  back to the very early 1900’s, and its still in good shape.  I think its worth a good bit of  money these days.
Women rode side-saddle for 1 reason, and for  1 reason only, because: a long, long time ago, women wore  skirts and dresses.
It was not a “choice” to ride side saddle… since women did not “choose” to wear skirts in the old west, any more than men of the old west  “chose” to wear pants.  That is just the way it was.
Now that women don’t wear skirts or dresses anymore, there is no “need” for a side saddle.  Its kinda pointless for a man or a woman to ride side saddle if they are wearing pants, just as it would be ridiculous for either a man or woman to ride astride in a skirt.
A side saddle is no big deal if all you are doing is traveling or going somewhere, its not all that different from riding astride, because if you are in a saddle, then  you  are in a saddle, it really doesnt make much difference what kind of saddle if you are just going somewhere or if you are on a trail ride, esp if your ride is on reasonably level ground.
On the other hand, if you are a working cowboy, roping, herding,  chasing steers with quick stops, fast starts, and sudden turns,   hunting up and down mountains,  then riding astride makes much more sense.  Its kinda hard to rope and tie calves in a side saddle. Women in the old west were not working cowboys - therefore, there was no need for women to use a western saddle.
People like Calamity Jane, who were mannish, were quite an oddity and quite rare in the old west.
I personally prefer riding bareback, and most/nearly all of my recreational riding is bareback. On the other hand, I would use a saddle for long trips, hunting ,  working, or for horses with sharp backs.  If I was to ride while wearing a skirt, I would use my side saddle.