On realms, when you open the player inventory on a horse, the first item in the inventory gets moved to the added lead slot on the horse and every item in the player inventory is visually moved one spot to the left. However, when you click on one of these items, it changes back to the item that was originally in that spot. Doesn't do this in single player, and still happens with no other mods installed. Also, you can place any item in the lead slot on realms.

Minecraft Wiki states that mods and plugins cannot be installed on realms.
The mod is not made with realms in mind and I'm not sure I can test the mod on realms somehow.

I only now learned what realms really is by reading the wiki (apart from knowing that it's related to multiplayer/servers somehow), and I play this game since 2010.

Disabling the slot in the server config didn't disable it on realms unfortunately. I'm not surprised it works client-side; I used to use the horse buffs mod on 1.21.1 and many of the features that are shared between the two mods were client-side on horse buffs.

You can use client-side mods on realms. The other client-side features of the mod seemed to work fine on realms, such as the horse not kicking you off in the water and visually lowering the horse's head. If you could add a setting in the mod to remove the added lead slot in the horse inventory gui, I think that would fix it well enough. If not, no worries.

It's surprising that it works only client-side at all.

The mod has a setting to remove the slot, but it's in server config (in 1.21.1+, 1.20 has it in common). You can try to disable it, but i'm not sure that it will work as it'll expect it from server most likely.