I coulda swore I did this already but maybe not.

As of mod version 1.1.0, the mod is not automatically detecting rideable modded animals and they have to be manually added to the configuration file. Such examples include the horses from Realistic Horse Genetics or the cows from Genetic Animals, though other players in the latter's discord says that it did it automatically for them.

Remaking the config file did nothing to fix the issue. Even now I had to go find the mob ID for the genetic horses to add cart functionality. I don't know what's causing it to not recognize rideable mobs.

I'm having this issue now as well, I'm not sure what is causing it as previously it was working fine, all the animals I had entered into the config were tested and worked. I've just now regenerated my config file in an attempt to fix it and the issue persists on both Singleplayer and Multiplayer.

My config had the following, any glaring errors you notice? Though it worked before so I'm not sure why it would suddenly stop working... is possible the keybind changed somehow?
pull_animals = ["horse_colors:horse_felinoid", "eanimod:enhanced_cow", "betteranimalsplus:reindeer", "eanimod:enhanced_llama", "eanimod:enhanced_mooshroom", "minecraft:skeleton_horse", "minecraft:zombie_horse", "eanimod:enhanced_moobloom"]

EDIT: Just checked my controls, it is still bound to "r" so not idea what is going on
EDIT2: Solved, the issue was Small Ships mod also using the "r" keybind.

Thanks for reporting!

I've tested with RHG and it works automatically for me via summoning horse_colors:horse_felinoid and taming it.