There is an incompatibility with Entity Texture Features, and specifically the wings part of Ears.
Having ETF installed will cause the wings not to render, and the pixels on the skin's alpha layer to render transparent. This issue persists even when disabling ETF's Player Rendering Features in the config.

Ears without ETF Mod: Wings render as normal

Ears with ETF mod: Wings do not render, parts of the player's head renders partially transparent.

Full Modlist for this demonstration:
3D Skin Layers (Was testing to see if this broke the mod earlier.)
Better Third Person (To see the skin easier)
Ears
ETF
Fabric API
Mod Menu

I have not tested this on Quilt, or Forge.

Having the same issue on fabric 1.20.1, parts of my skin are transparent and wings are not visible.

I experienced this problem too
but I figured out it was the setting accessed via the resource packs menu
"Better vanilla skin transparency"
If you disable that, they should reappear
the random transparency though? I haven't noticed that yet..

Ears sequesters additional data that there's no room for anywhere else by hiding it in the alpha channel of the parts of the skin that vanilla forces to be full opacity. This includes the wing and the cape textures.

With the current way that the Ears skin load patch is structured in most versions, another mod interrupting this loading to keep the alpha will break the loading of this data. A fallback path needs to get added that will load the alpha later and then set it to be fully opaque if Alfalfa data was detected.

In the meantime, yes, the workaround is to disable "Better Vanilla Skin Transparency" in ETF.