Side-saddle riding died out in the early 20th century due to what would, no doubt, be regarded by some to be the start of the feminist movement. Before this time, for a lady to show even so much as her ankle in public (clothed or otherwise) would have been looked on as both vulgar and forward behaviour. Therefore this, together with the certain delicacy (or rather, indelicacy) of the position, prohibited women from riding astride. How and when it actually started, I know not, but at one point, no doubt, some enterprising and bold lady decided that she was going to ride like the men. Naturally, it instantly became the fashion, and thus, side-saddles were cast aside. The change in style had nothing to do with danger, for just as many men broke limbs out hunting - probably more, for women on the hunting field were, in fact, a rarity.
Since this time, fashion has turned full circle, and side-saddle riding is now on the increase. It is great fun, quite as safe as astride for both horse and rider (we haven't lost a member yet) and just about the most elegant way to ride. Nor is it exclusively for the ladies. Male riders are often intrigued enough to 'have a go' - and manage creditably.
Far from being outmoded, it is actually very much de rigueur.
Loxwood, West Sussex