Photos Library error 5416

If that's not it, then try starting in Safe Mode (this is different for different computers)

           Start up your Mac in safe mode - Apple Support 

Safe Mode bypasses certain potentially disruptive processes, including automatic login items.

Also, how are your preferences set, especially:

In Photos>Preferences>General,

Is "Use as System Photo Library" grayed out (indicating it is) or solid (indicating this isn't your system library)?

Is "Copy Item to the Photos Library" checked? (This is the default, indicating a "Managed Library")

In Photos>Preferences>iCloud

Is "iCloud Photos" checked?

If it is, is "Optimize Mac Storage" check?

It's good that you have tried repairing the library-- that's always a first step. (Well, after restarting the computer, of course.)

I'm worried that you might have your library stored on an external drive that is not formatted for Mac. If a Photos Library is stored on an external drive it must be formatted in either APFS format or in Mac OS Extended (Journaled) format. If you're using exFAT or some other format, you can't get full functionality, and can even damage the connections needed for the Photos database.

If it's not the hard drive, see my next post: