Guess it reduces the chance of falling off to the right, with no leaping horn to hold you in. But that would counter the reason for NOT having the leaping horn, easier, FASTER to get off the sidesaddle, should your horse go down. American ladies didn’t always ride groomed trails and roads, so rough ground did make chance for horse tripping rather common. With the leaping horn, you are pretty locked in place, hard to get off quickly should a horse fall.
I wouldn’t ride in a sidesaddle with that rail, don’t like it. Bet even WITH heavy clothing, if you got bounced in the saddle, landing on that rail would be EXTREMELY painful.
I expect it is one of MANY adaptations to saddles, trying to improve them, make them safer for riding, though this rail looks like a plan that wasn’t well thought out. Those reproductions of Sears catalog show a huge variety of saddle styles to choose from in sidesaddles.