The bug

Horses do not make step_wood sounds when walking on Nether wood, bamboo wood, or stems.

To reproduce

Put a horse on a platform made of different types of wood. Lead it around with a golden apple (not while riding it) and observe that only the Overworld wood types (those with SoundType.WOOD) will produce the intended sounds.

Solution

Also check for bamboo_wood, nether_wood, cherry_wood, and stem sound types when checking for the wood sound type in AbstractHorse#playStepSound(...).