Just placed KuzMax.png into .minecraft\CustomSkinLoader\LocalSkin\skins, and instead of local skin, I still see my Mojang's one.

My player's username is KuzMax.

Attaching log:
CustomSkinLoader.log

Regarding skin overlays, the problem is not related to this mod.

I finally managed to get this work, but I had to remove everything in CustomSkinLoader.json loadList, except LocalSkin entry. I think something's wrong with the loading priority.

Mod's version: latest current
Fabric version: latest current (fresh installed)
Minecraft version: 1.15.2

Moreover, top layers are not loading at all, so instead of this:

I see this:

(without mod there's no such an issue)

Skin from Mojang is loaded in highest priority. You can modify loadList.

It's a good idea to mention this in docs under LocalSkin section, because it confuses users a lot