That was the short version of the story. Getting there wasn't as easy as it sounds because hustior failed to mount the root filesystem after it had restarted in a new namespace. Some tests later I've learned that even trying to access any file that doesn't belong to your user fails. So I've tried some unshare commands from Owncloud in a container: container and namespaces but they failed too. This failing led me to https://github.com/canonical/lxd/issues/12882 and the following workaround:
echo 0 > /proc/sys/kernel/apparmor_restrict_unprivileged_userns
echo 0 > /proc/sys/kernel/apparmor_restrict_unprivileged_unconfined
After this hustior worked again.
Searching for apparmor_restrict_unprivileged_userns brought me back to the article about the restriction of the unprivileged user namespaces. I've had it found previously but as I was able to start hustior in a new namespace I thought it didn't apply here. Turned out I was wrong about this.
As I didn't want to disable security features I've read the article again and found out that the simple template is all that is needed. Now hustior works with Ubuntu 24.04 too.