Photo Library - Reinstall from Time Machine

I recovered the library from Time Machine and when I open the Photos app it says "The library "Photos Library" has inconsistencies that must be repaired.

How did you "recover from Time Machine" ? Did you launch Time Machine and restored the library from the Star Wars display? And did you restore it to your internal drive or an external drive?

If the library is on an external drive, how is the drive connected and how is it formatted?

Photos has only this one Repair option. If that does not help, there is not much you can do to rescue the restored library. You could try to restore an earlier version, in the hope, that this version will be better.

All of my photos/video are in sync via icloud.com and the photos app on my phone, but the app on my Mac is now not working and not being synced with the others.

If all your photos are in iCloud, you could try to start with a new, empty Photos library and sync this library with iCloud. Photos will download all photos and albums, but the Faces data will be missing. DO you have a fast internet so this would be feasible for you?

This document describes what will be restored from iCloud to your Mac. It will be just like syncing to a different mac, see: Use Photos and iCloud Photo Library on multiple Mac computers - Apple Support

Once I open the Photos App, I get the Repair Library warning stating The library "Photos Library" has inconsistencies that must be repaired.

Even, when you hold down the alt/options key while launching Photos, and keep the key held down until you are seeing the Library Chooser panel?

This should let you create a new library.

Will this "new" library override the existing one?

In the Library Chooser panel click "Create New".

Pick a different name for the new library, so your existing library will be saved.

It just creates a new one?

Yes.

And then it should automatically populate based on what I have via the cloud? Thanks for all of your help with this!

Once you see that the new, empty library works well, open the Preferences in Photos, and click the button "Use as System Photo Library" on the "General" tab of the preferences panel. Then enable iCloud Photo Library for this library in the iCloud pane of the preferences.

Only make sure, you have enough storage on your drive. You may need to move the old library to an external drive, if storage is an issue.

Yes, it is only the options key (on some keyboards called alt key), that will bring up the library chooser.

If you use the command/alt options combo, you will launch Photos into the Repair mode, that you are seeing all along.

You're welcome. Good Luck.

You can hide the Photos window, but I would not quit Photos. In the Photos > Preferences panel is a button "Pause for one day". You can click this, if you need your network for other purposes.

Do you have the Photos > Preference "Optimise Mac Storage" enabled? If you have enough free storage disable this. Then iCloud will download the original versions and not just optimised previews.