All my (downloaded?) files restored from Time Machine are marked as "damaged". I assume the problem is similar to this one: All downloaded files cannot be opened as they are 'damaged and should be moved to the Trash' so I try to change the "Allow app..." setting.
I open Security & Privacy and see App Store is selected:

I click the padlock and check "App Store and identified developers":
(I also notice that "Anywhere" is no longer an option.)
I leave System Preferences and try to open my files, but still get the error. So I go back to System Preferences, Security & Privacy and the old option is still selected:
