It's easy (if not pleasant) to clean up a dog's poo - a plastic bag, scoop, twist, tie, pack in bag, and it's done.
Some horses stop to poo (not trained to keep going). A single rider could dismount to collect in a container, and then load it in a saddle bag, He or she might need to do this anywhere between one and four times depending on the length of the ride.
However, most horses poo while in motion - which means the owner would have to dismount and walk back, picking up each of about 20 pieces. The distance would depend on the speed of travel. A horse's walk averages 6 kph, the trot 14 kph, and so on. At a gallop, 30 kph is not unusual. A minimum comfortable distance for a ride and a decent amount of exercise for basic health and fitness is is about 10 k's. Twenty - 40 ks if training for endurance or marathons.
If you have a group of people out riding, this means the entire group must stop for every poo event of every horse - hence destroying the enjoyment of the ride for everyone - making it pointless going out in the first place.
Dog's poo has a foul smell if fed commercial products rather than fresh meat and veg. An absent-minded person can easily step in it. If that happens it sticks to the shoe, is hard to remove and stinks out the car on the way home.
Dogs will sometimes eat other dog's poo. This means the risk of picking up worms or other diseases.
Horse poo has a pleasant smell. It is usually large enough to easily see and avoid. If one steps in it accidentally, it wipes off easily on a bit of grass, dirt or stone. If one gets in the car with a smeared boot, the smell is barely noticeable even for sensitive noses.
However, there is an invention - a kind of open curved catchment bag, that riders can attach from the saddle, under the tail, curve under between the hind legs, and attach to the girth. It can catch both wee and poo.
It takes a lot to train a horse to accept it. It would be beyond the training skills of the average amateur week-end rider.
And at present it is considered so laughably ugly that I've never seen anyone actually using it. Like many matters of appearances, it would take widespread acceptance before people would not feel too acutely embarrassed to try it.
Perhaps one solution would to be to create separate trails for walkers, cyclists and riders.
