The elytra texture does not get displayed right in any way.

If I only wear an elytra parts of my skin get displayed on it.
If I wear an elytra an any other armor part parts of the armor part get displayed.
If I wear an elytra and I hold anything in my hand a grid of textures gets displayed on the elytra.

To recreate simply put on an elytra and do one of the upper things.