I have tried to restore the library, make a new one, copy it, relink it and it appears damaged again and again, and it does not save any backups. I can open it and work but every time I open it again I get the damaged message without continuing to make backups.
Try to export a library XML, create a new blank library, then load that XML. Note: FCP libraries should only be placed on a local disk volume formatted as APFS or HFS+, not ExFAT, not on a network server, nor a cloud store.
Internally, the FCP library consists of several SQLite databases, each of which contains multiple SQL tables. From a filesystem level, each one is a file named something like CurrentVersion.fcpevent. If one or more of those databases are missing or damaged, FCP, upon library initialization, will produce the "appears to be damaged" error.
The backups should always be available. By default, they are stored in ~/Movies/Final Cut Backups.localized. It typically keeps many backup versions of each library. If you examine that folder, are they not there? What is the last modified date of each one? As a precautionary step, I suggest copying all the backup libraries to another hard drive.
FCPX 11.1 don't save backups and the library appears damaged
