I've released the Shoelace theme version 18.104.22.168 for Moodle 3.4 with FontAwesome 5: https://moodle.org/plugins/pluginversion.php?id=16031.
It contains the 'free' version, so all current 929 icons: https://fontawesome.com/icons?d=gallery&m=free. The theme is 'beta' and I noticed that theme switching via the URL does not work completely and you need to do a 'Purge all caches' but should be fine after that / set as the main theme. I suspect down to the core icon system code.
All being well I intend to add the same to Essential, but that will take a little time as I'll need to write an update script to convert the icons in the settings from FA4 to 5. But as this proves this can be done then really core Moodle should update too.