Services like systemd-journald and systemd-logind show a failed state with error code 228/SECCOMP, and return an error message when attempted to be started using systemctl. This also caused journald logs to not be created or updated for any services.
The problem on my server was that the kernel I am using seems not to support seccomp, which is required by systemd 231+. See https://bugs.debian.org/cgi-bin/bugreport.cgi?bug=832713.
Edit the corresponding systemd unit files (e.g. /usr/lib/systemd/system/systemd-journald.service) and comment out the MemoryDenyWriteExecute and SystemCallFilter lines.
To make the change persistent across updates, copy the unit file to /etc/systemd/system/ and make the changes there.
systemctl daemon-reload and restart the failed services, or reboot.