I’m using Apple macOS Ventura 13.2.1, Publii Version: 0.41.1 (build 15919).
What I’ve found is that when changing the theme first time as you describe, the theme does indeed change, but we must first close Publii and restart. Then the Site Settings shows the new theme is shown. Plus of course you can verify with Preview.
If I then change the theme back to the original, when we choose Save, a popup message confirms the theme has been changed back.
Also, sometimes, when you change the theme after the first time change, the Save button light blue background colour changes grey perhaps to indicate confirmation of change.
So on macOS at least, for completeness, it looks like this latest version of Publii is missing a minor popup message indicator, and perhaps a prompt to tell the user to close and restart Publii.
For now, the key thing is to close and restart Publii, then check the Site Settings to confirm you change has been made.
If you’re using Publii under Microsoft Windows, do you find it’s any different?