When right-clicking untamed horses, donkeys, or mules while holding an item, the hand animation still plays even when the mob is on cooldown

Following the resolution of MC-100258, untamed horses, donkeys, and mules now have a cooldown period when making them neigh by right-clicking while holding an item. Right-clicking them not only triggers sound effects but also causes them to rear. However, when attempting to do so while they are on cooldown, this action does not occur. Nevertheless, the hand animation associated with right-clicking still plays, despite no action taking place.

Steps to reproduce

1. Summon an untamed horse, donkey, or mule
2. Continuously right-click it while holding an item

Observation: Even when on cooldown, it continues to play a hand animation.
Expectation: Right-clicking while on cooldown should not trigger a hand animation.