Ok, I solved the issue. The documentation is missing a critical step. You first need to run:
sudo graylog-sidecar -service install
The systemd unit file gets created, and then systemctl works as expected.
Found the answer thanks to pointers in a few previous topics:
https://community.graylog.org/t/installing-graylog-sidecar-on-linux-2-x-rhel-6/
https://community.graylog.org/t/unit-graylog-sidecar-service-not-found-ubuntu-linux-20-04