Windows10 v22H2 OS Build 19045.2728 per WinVersion
Edge v111.0.1661.44 (Official build) (64-bit)
I use the modified shortcut to launch Edge:
"C:\Program Files (x86)\Microsoft\Edge\Application\msedge.exe" -disable-windows10-custom-titlebar
I don't think I've seen this mentioned. I have discovered that if I close Edge using the Window Close Button in the upper right, the next time I launch Edge using the shortcut it does not skin. If I close Edge using the Edge Menu "Close Microsoft Edge" command, which I typically use, the next time I launch Edge using the shortcut it does skin.
I've checked this repeatedly with a few different skins, including Corporate and Diamond, and it has been completely consistent. I don't know how long it may have been this way.
It does not seem to matter with Chrome.
Thought it worth passing along. Thanks.