Ooooh, sorry. We really need someone who knows MySQL and Certificate to answer your question (this is not me).
Just to be clear: you go to your course. Before you uninstalled the certificate, it had a certificate in it. And after you uninstalled and reinstalled the certificate module, the certificate is no longer there. Is this correct?
I have a this idea that there is another MySQL database table involved that you did not backup, thus it does not have a record that a certificate is there. I bet if you had the My Certificate block (report) installed, it might show the certificates to the administrator.
I do know know what happens to certificates that maybe stored in /moodledata in each course folder, when certificate is uninstalled.
If it was me, I would create a localhost and install certificate, create a course and add a certificate. Have a student receive a certificate. Then look at the MySQL tables and moodledata. Finally, uninstall and reinstall certificate and see what didn't happen.
I suspect that Moodle code picks up on the fact that Certificate has been installed. When you add a certificate to a course, it add a record to one of the course MySQL tables which has a field that points to a record in a certificate table. Maybe that link is no long valid, or uninstalling certificate automatically deletes the link.
Hope somebody else steps in. Chris
