In addition to Spotlight, have you tried a Finder search? Search for ".photoslibrary"
The result will give you the path.
Be sure to check the trash. Also check iCloud Drive or Mobile Documents. The Library doesn't belong there at all.
Are you backing up the Library itself. And how is your external drive formatted? To avoid damaging the Photos Library an external drive must be formatted in either APFS format or Mac OS Extended (Journaled) format. Additionally, the drive can not have had Time Machine on it since it was formatted. There have been so many problems with using incompatible drives that the newest macOSs won't even allow a Library on a non-Mac formatted drive to open, since there is a chance of damaging the Photos database. See this:
Move your Photos library to save space on your Mac - Apple Support