Unity's doc page on WebGL Templates is rather straight-forward, but the template does not have the full-window mode. Thanks to the author of the repo from which I forked, the neat index and styles work as they are supposed to. The subfolders are the branch names. There could be useful troubleshooting workarounds, I did not have the time to look closer at them and try in different platrofms/browsers. Updated the Unity Editors versions and everything works well.