Can the database backup tell me which add-ons were installed
Most add-ons create a table named something close to its actual name.
For example our Bootstrap Image Gallery add-on creates a table named "btBootstrapImageGallery"
It may also be helpful to install a clean version of concrete5 somewhere else so you can see what a clean database looks like before any add-ons have been installed.
When comparing, if you see a table name that does not exist in the clean concrete5 database, and the name is missing from the /packages directory, this is more then likely the missing add-on.