Versions:

• Curious Elytra: Forge 1.16.5, version 4.0.3.5
• Caelus: Forge 1.16.5, version 2.1.3.0
• Curios: Forge 1.16.5, version 4.0.5.0
• Forge/Fabric: Forge
  Observed Behavior: I stopped my server to add Curious Elytra after discovering that my newly crafted Silent Gear elytra could be equipped to a back slot. Upon starting my server again, I could not connect. My brother, who didn't have his elytra already equipped, was able to join. However, when he equipped his elytra, it crashed his client.
  Expected Behavior: Happily flying away with my elytra.
  Steps to Reproduce:

1. Use Silent Gear.
2. Craft an elytra using the SGear blueprint (I used only phantom membrane; idk if that is important).
3. Use the above versions of Curious Elytra, Curios, and Caelus, as well as Silent Gear 2.6.0+217.
4. Equip elytra in back slot.
5. (not sure if these steps will reproduce the problem exactly, but these are the only steps I did other than already having other mods installed, which you can see in the crash log).
   ...

Crash Log: https://pastebin.com/4SRyjuKQ

You listed your Caelus version as 2.1.3.0, however the one you actually have loaded in the crash log is 2.1.2.2.

Update Caelus to 2.1.3.0 and that should fix your issue.