So, I finally fixed my problem. I believe that the problem was introduced when Apple made its TV app an integral part of macOS back in Sonoma. The solution, though simple, can require some patience.
First, find TV app in macOS. Start it, navigate to the "Account" menu, scroll down to "Authorizations" and click "Authorize This Computer". Likely macOS will tell you "This computer had previously been authorized; it has been re-authorized."
Then, move back to "General -> Sharing". Uncheck the check mark on "Media Sharing" if it is already checked. Then, use the ⓘ next to "Media Sharing" to pop up the sub menu. Make sure "Share photos with Apple TV" is checked. (If you don't want to share all content, click "Choose" to make selections.) Click "Done" to return to the previous menu. Then, check "Media Sharing" again. macOS will ask you to enter Apple account password. Confirm. macOS should pop a confirmation to say "Media Sharing is now On." After this, go into ⓘ again to see if sharing is really on.
Go back to Apple TV's Computers app. Click the one that you just shared. If the library loads, great. If the library is listed, but Apple TV still says you don't have library shared, blah blah, open Settings to Users and Accounts. You need to log out of of sharing account. Assuming that your home sharing is using the default user's account, move cursor to "Default User", scroll down to "Remove User from Apple TV". Confirm. Afterward, add your user back and reenable Home Sharing on Apple TV to use this sharing account.
Start Computers app again and see if shared library becomes available. If it is still not available, go back to your Mac, start macOS's TV app and reauthorize the computer again. After this, check ⓘ next to "Media Sharing" again to make sure everything is good. This should solve the problem.
- So, what is the trouble? Here are my speculations.
Before integration of TV app into macOS, Home Sharing only requires the tvOS and macOS share the same Apple ID. If you authorize macOS for Home Sharing, tvOS can access its media library. Since then, media access control has evolved. When macOS integrated TV app, media sharing permissions is transferred to TV app for some reason, and not the Photos app (formerly iPhotos) whose library is being shared.
If you are like me who started using macOS before Sonoma, you probably never started TV app on a laptop. Even after you have started TV app, you probably have not bothered to find and click that "Authorize this computer" button. In my opinion, that you have to click that button in TV app (at least in Sequoia, perhaps in Sonoma, too) is a bug. Your macOS is configured to the Apple ID; any subscription/purchase you made is already authorized to play on you Mac. But the Mac is not synchronized with Apple TV's play history and so on, and Apple TV cannot access the Mac's media library.
Another possibility is that you have too many "authorized" devices because you have been using Apple devices. Unless you subscribe to iCloud - and I don't, you can only authorize five (5) devices to share one Apple ID. (I have no idea whether a paid iCloud subscription will allow you to authorize more than five.) When your older devices die or being replaced, they are not automatically deauthorized. So, you have to actively authorize your new Mac and bump out an older device.
So, I also recommend that you go to "Account -> Account Settings", find "Manage Devices", then remove any older devices that are no longer in use. (You can do this in either Music app or TV app.) This should not have anything to do with Home Sharing, which shares your own photos, not anything Apple owns. But that seems to be how Apple messes up these features without adequately testing various device migration paths. Even after I initially succeeded in sharing my photos to Apple TV, I still found some previously working functions broken, including pairing of my AirPods.
