The systemd.service manual page recommends using "Type=simple" whenever possible. The systemd service manager works best with processes that do not forks and exits in the parent but rather just runs. By adding -D on the ExecStart line in ssh-agent-service ssh-agent will do this, and the type of the service can be changed to "simple".
Sorry, it doesn't look like a good universal solution.
I was wrong closing this bug. Please apologize me. At least Debian runs ssh-agent with -D so we also need to consider this decision. Thank you for bringing it and thanks to Jakub for fixing my fault.
This bug appears to have been reported against 'rawhide' during the Fedora Linux 38 development cycle. Changing version to 38.
FEDORA-2023-64f8335634 has been submitted as an update to Fedora 39. https://bodhi.fedoraproject.org/updates/FEDORA-2023-64f8335634
FEDORA-2023-64f8335634 has been pushed to the Fedora 39 stable repository. If problem still persists, please make note of it in this bug report.