The first is that carnivores large enough to ride are few and all are very dangerous. Grizzly bear, Siberian tiger, Lions all are large enough for a human to ride. However all are dangerous, even today when these are made pets they still can maul people, even their owners.
Second, especially for times past, these animals are a large competitor for prey animals and require a lot of meat to keep happy. You don't want your ride to decide that you will make a nice snack. Keeping a large lion fed is very expensive. That is why so many are in shelters or have to be put down, they cost too much to keep, and that is as a pet, not a hard working mount.
Many of these predators are also solitary, (lions are pack) partly because it is easier to feed fewer mouths when you get bigger. Wolves are pack animals and a human can replace a wolf as the leader. House cats are still partially domesticated.
Most animals we ride are herd animals, they are easy to feed, usually grass and other vegetation (often stuff we can't digest well) so there is no competition for food and they are generally not going to try and kill each other.