I enabled tmpfs overlayroot running following.
$ sudo sed -i 's/overlayroot=""/overlayroot="tmpfs"/g' /etc/overlayroot.conf
After rebooting, Ubuntu boots as read-only and everything works as expected and nothing is written to disk (confirmed using disk checksum).
Then, when I set disk to hardware read-only (a feature of datAshur PRO² USB flash drives), Ubuntu boots to emergency mode.
UPDATE: using fsck.mode=skip
solves the issue but why do I have to skip filesystem check?
Why? What am I missing? Thanks for helping out!