In reality, what Chromebooks provide is a reinvention of the good old mainframe and terminal principle. In theory (like my recent - half joking - 9front comment) this is something that would be really easy to set up with nearly all Linux systems and especially immutable ones.

My take would be:

Put an sign up / sign in form as a “first boot” message in a distribution of your choice where you can specify (or have pre-filled by an organisation) a central server (could be something fancy like Nextcloud or something simple rsync based) where your whole profile folder gets synced to. After that: If anything goes kaputt just roll back the sync. Or “powerwasch” (to keep the ChromeOS terminology) the system to a clean state and re-sync your home folder.

In theory something that could all be implemented with a little scripting in an afternoon.