Thanks for sending the reproduction project.
You need to add a little bit of Z offset at the outline-only GameObject named GameObject in your example scene, otherwise Unity will sort it to be rendered after (and thus on top of) the main skeleton mesh in your case. So set the GameObject's Transform Position Z value to e.g. 0.01.
BTW: in spine-unity 4.1, we have added a RenderCombinedMesh component for the purpose of further improving support for rendering outlines with multiple-submesh skeletons, see the FAQ section here under "The Outline shader displays inner outlines around skeleton parts. Why?".
Extra: Just noticed the Spine examples of the outline shaders don't work on my project or the new one either... They don't look like these ones.
Spine URP shaders are separate from Built-In Render Pipeline shaders. This also applies to outline shaders. The Outline Shader URP example scene is located here:
com.esotericsoftware.spine.URP-shaders/Examples/Outline Shaders URP
Likely you had a look at the Built-In Render Pipeline Outline Shaders example scene, which like all scenes using special Built-In RP shaders will display incorrectly with URP.
