I'm using a modpack on CurseForge for Minecraft Version 1.19.2 / Forge 43.4.6 and it includes Domestication Innovation 1.6.1.

I tamed a horse and equipped a saddle before using a Rotten Apple to turn it into a Zombie Horse. The Zombie Horse has different stats from the original horse, but the saddle disappeared and it was no longer owned by me. I'm not able to ride the Zombie Horse or equip a saddle. I can feed the horse the Sinister Carrot to turn it into a Skeleton Horse, but the Skeleton Horse has the same restrictions as the Zombie Horse where I can't attempt to ride it or re-equip the saddle. I've tested this out on a lot of horses in a creative mode world, and all the transformed Zombie and Skellie Horses have identical jump height and run speed.

I have encountered regular vanilla Skeleton Horses and used Create Mod to transform tamed horses into Skeleton Horses with this modpack, and they behave as expected. This issue seems limited to the Zombie Horses and Skeleton Horses created from the Rotten Apple and Sinister Carrot transformations.