Photos 11 in Tahoe won't open older Photo Libraries

You were close with the option key, but the steps are a little different.

First, quit from the Photos app entirely with Command-Q or the Quit command in the Photos Menu.

Next, hold down the option key while launching the Photos application from the Dock or the Applications folder.

Photos should display a list of all of the library files it can find on your disk volume(s) that it thinks it can open, with indications of which is your current system library and perhaps other status notes.

Select the library you wish to open.

Later, you can repeat the process to switch back to your normal library. Until you do that, Photos will continue to use whichever library you opened last.

"The photo Repair option, selected with the Option key + Command key and double clicking the Photos App, seems to only work on the default library."

When you double click the Photos.app icon, this will open the last Photos Library you have used with Photos, not necessarily the System Photos Library. To repair a different Photos Library, not the last opened one, you will have to use these steps:

• Quit Photos, if it is running.
• Hold down the key combination ⌥⌘ firmly while double clicking the library you want to repair and keep holding down the keys.
• Release the key combination only once you are seeing the dialog asking, if you want to repair the library.

Is your old library perhaps an iPhoto Library and not a Photos Library? If the old libraries are going back twenty years, they might not yet have been migrated to Photos. The last system version being able to repair or migrate iPhoto Libraries has been macOS 12 Monterey. Since then, Photos for Mac can import photos from iPhoto Libraries, but no longer open them for browsing and create Photos Libraries from them.

Bounder6733 wrote: …It does reside on an external hard disk, but it's directly connected, not through a network or cloud.

We should ask: 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. 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

If this drive is in a an incompatible format, stop running Photos with it immediately!  A Photos Library can sit on an incompatible drive, but running it may corrupt the database.

You can see the Libraries that Photos knows about by closing Photos and option clicking the Photos icon to get this dialog:

This shows the Libraries and their paths, and it indicates which Library is the System Library.

If you get to the point where you want to merge some of the libraries together consider the paid version of PowerPhotos. It does a better job of merging/importing one library into another than Photos.

I ran tests and posted them in this user tip: Merging/Importing iPhoto/Photos library into Another Photos Library: PowerPhotos vs Photos.

If you quit Photos you can just double click on the library of choice to open it. You don't need to go thru the Open pane if you know where the other library is located.

Also if you try to open a library when another library is open you'll get this window;

Most of the time if you click on Switch it will close the current library and open the new one. There are times when it just closes the current library.

This is with Sequoia 15.7.2 on a Mac Mini M4. I'm assuming it's the same with Tahoe.