The icons in the builder interface are not powered by Font Awesome but are SVGs. Someone or some software might have added something in the WordPress
.htaccess file that prevents SVGs from being displayed.
With that said, please try resetting your htaccess file by renaming it to
.htaccess-bak. Then in WP Admin Menu, go to Settings > Permalinks and just click the Save Changes button so that WordPress will generate a new htaccess.
If that doesn't help, please remove everything including WordPress and do a clean install. You can also try copying the working sites for further testing.