“I” found a fix to this:
Plugin and settings: installed Elementor’s Cookiez plugin on the free plan. It’s active, set to “opt-in”, shown worldwide, with banner display on. Nothing is wrong there.
Your Chrome (logged in): there’s no banner because you’ve already accepted, so the site stores your choice. The small cookie icon at bottom left is the banner’s “change my settings” button. That’s probably why it looked fine when tested.
A fresh browser (logged out): no banner, and the console shows an error. LiteSpeed has minified the banner’s script and moved it into /wp-content/litespeed/js/. The script then looks for its language file (lang-en.js) in that same folder and gets a 404, so the banner fails silently. The file does exist in the plugin’s own folder (/wp-content/plugins/cookiez/assets/build/lang-en.js), and that address returns 200.
Go to LiteSpeed Cache → Page Optimization → Tuning → JS Excludes, add cookiez on a new line, and save.
Go to LiteSpeed Cache → Toolbox → Purge All.
Check the site in a private window. The banner should now appear.