Many movies missing when using Home Sharing

Our Movie/Music library is on our iMac, running Catalina (latest OS update). We have two Apple TVs (4th generation) that connect to our Library via Home Sharing (they're running the latest TvOS update). The Home Sharing connection is fine and we can browse the Movies, TV Shows, and photos using the Computers app on the Apple TVs. Unfortunately, there are several movies and a few TV shows missing from the lists. When we browse the library from the iMac, all the movies and TV shows show up (and play).

As a part of troubleshooting this problem, we've:

• Deauthorized and reauthorized at the iMac in both the TV and Music app.
• Disabled and re-enabled Home Sharing on our Apple TVs.
• Renamed the library in the Media Sharing settings in System Preferences.
• Attempted to leverage home sharing from an iPhone and iPad (they each see different subsets of movies, but never all of them, seriously wonky).

I've tried to find a pattern in which movies and TV shows are missing on devices, but there's not an obvious one. I thought it could be that the missing movies were purchased on the Apple TV instead of from the iMac, but all of the purchased movies (regardless of device that made the purchase) have been downloaded to the iMac. There is also no correlation between ripped DVDs and purchased movies versus the movies that are missing. Naturally, some of these movies are digital copies that came with DVD purchases, but all of our oddball accounts for the digital copies are linked to our iTunes account, and I haven't been able to find a correlation between the missing movies and a specific service.

One extra factor in our installation is that our Music and Movie libraries are stored on an external drive connected to the iMac via USB. The advanced settings in the Music and TV apps control the path to the media files. We've used this setup for several years and these paths were migrated forward with the Catalina upgrade. It's been working fine for months and only started having trouble in the last few weeks (perhaps associated with a recent system update?).

I don't know what else to do to test/fix this. Suggestions are appreciated.