So I'm rewatching the entire LoTR trilogy, like you do, and I had a thought: cavalry charges historically don't work, but are really really cool looking, maybe a Legendary trained horse could do it. What if animal training skill offered additional benefits to animals your civilization has already fully domesticated, such as greatly improved animal performance? This would essentially offer additional ranks beyond fully domesticated to make it worthwhile to continue using the animal training skill even if you've killed or domesticated all the animals that appear on a given map. Training should increase the attributes and skills of animals in some way so that novice animal trainers that can barely keep an animal tame also make subpar hunting and war dogs, and subpar mounts whenever they're working in game.
The power goals have stuff like tripping over stones and dogs that are specially bread to track and kill werewolves. Both of those sound like the kind of stuff that could be incorporated into animal training to some extent. Higher skills make more powerful war animals, sneakier hunters, and mounts that are more capable of keeping their footing with a rider or in battle. Animal training skill could determine whether a horse refuses to rides towards armed attackers and tosses its rider in difficult terrain, or whether horses are capable of amazingly charging down a horde of armed goblins and without tripping over them or being immediately stabbed to death. Dogs trained by legendary trainers could have increased speed to keep pace with their dwarves, or do more damage. War animals should also become somewhat larger when trained and fed by skilled trainers, so several generations of dogs trained by a legendary trainer are of exceptional size and strength for their species and potentially becoming a distinct breed.
Eventually advancements in animal training should impact a civilization, so it's easier to generalize for world generation. Elves train mounts capable of fighting simultaneously with them, or Goblins become so good with giant bats they develop flying mounts (which they have now of course, but in theory dwarves would also be able to gradually have these kinds of things without anyone having them by default). This means somewhere along the line a civilization will develop horses that can shove over a bunch of armored fighters and overrun them without tripping, but with any less than legendary +10 horses they'll probably end up realistically skewered.