As a hobbyist horseback rider, I can tell you what happened to people who genuinely love to ride horses: they became a hobby accessible to those with enough interest and money, and more common in rural areas (since as other comments have pointed out, horses - like cars - make much less sense in dense urban areas).

In my part of the rural US, one can get an isolated hour of horseback riding for $20 or so, or maintain a personal horse for a few hundred a month (if kept at a stable run as a business for that purpose; cheaper if you have your own property suitable for housing the horse).

I can see car garages and designated driving areas becoming the riding stables of the future.