I've been watching very old street scene footage from the 1890s-1910s. Something I keep noticing is that even on streets packed with horse-drawn carriages I never see someone just riding on horseback. It seems like horseback riding would be more efficient for someone wealthy enough to afford their own horse. Was it not allowed? If not, what was the rationale?
Why do you drive a car instead of ride a motorcycle?
Well, those same reasons largely apply to why a carriage was more popular than riding astride. Riding a horse is a lot of work, it's risky, it's tiring, it's more work for the horse, you aren't shielded from the elements, and you can't carry nearly as much luggage. If you and your family are riding, it will take a horse you, your spouse, and each of your children. You all have to have appropriate horses, tack, and the skills to do so safely. Alternatively, you can have a team of two cart horses driven by an experienced handler, and you get the added benefit of actually socializing with your company.
As for your question, I can't find any specific laws forbidding riding astride. There have been regulations placed on how fast one may ride, but never that you had to dismount and walk.