After spending a considerable amount of time going through my 91 mods, I have identified the culprit behind the issue I am facing. It appears that the MorePlayerModels mod is causing the problem. Although it does not cause any crashes, the elytra do not show up on the player's back. Even though the custom-made elytra can be worn, they do not appear on the back and the player appears to be lying down while flying around with no elytra.

What version of the mod are you using?