I expected that right clicking Elytra would behave like right clicking a chestplate and equip it.
What happened was that right clicking it adds one on my chestplate slot, but does not remove the one in my hotbar.
to reproduce, right click an elytra when you have no armor on.
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yes, was adding it to a map.

That is intended. It doesn't do work in survival. Try doing it with any other equippable item in creative.
Cannot Reproduce. Are you in creative mode?