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
Tame a horse
Equip it with horse armor
Equip it with a saddle by right-clicking the horse
→ ❌ The horse armor disappearsUnequip and equip the horse armor again
→ ✔ Both the horse armor and the saddle show
Still present in 20w20b. I'd like to add that the armor re-appears when leaving/re-joining the world on Singleplayer, or simply removing and re-equipping the armor from the horse's inventory.