Horse armour stops rendering if you equip the horse with a saddle by right-clicking

The bug

If you have a horse wearing armour and no saddle, right clicking with the saddle will place the saddle on the horse but will cause the armour to become invisible. It does not do this when you equip the saddle in the inventory, only when you right click with it.

To reproduce

1. Tame a horse

2. Equip it with horse armor

3. Equip it with a saddle by right-clicking the horse
   → ❌ The horse armor disappears

4. Unequip and equip the horse armor again
   → ✔ Both the horse armor and the saddle show