vazandrew - I am actually having the same error... and it is recent and since upgrade Catalina on iMac, to iOS 13.3.1 on iPhone X, and tvOS 13.4 on Apple TV 4th Gen. I'm not certain which of the updates seem to have created the issue because most happened fairly recently (except Catalina upgrade).
I have iMac 5K 27" 2017 running Catalina 10.15.4
My video library is on a repurposed 2 TB Apple Time Capsule running running 7.8.1 that I am using as a home media server (for several years)
Fully wired ethernet home network.... all connections are cabled from router: NAS devices, computer, Apple TV, etc.
On my iMac, I can load and play both purchased Apple content and Home Library video content.
From this iMac, I can launch a movie and then select to have it play / display on 2 Apple TV's that I have, again connected via ethernet.
If I go to the Apple TV (which has and has always had Home Sharing enabled properly) I can launch the "Computers" App, and can see my entire video library (over 900 movies). I can load and display the data about the movie but when I press play - I get an error saying that there is an error and the content cannot be loaded.
This same Apple TV will play all Apple purchased content and content from all other installed applications (Netflix, CNNgo, CBS, etc.) without any issues or problems.
All devices have latest software updates.
I have refreshed the Home Sharing settings on Apple TV.
I have removed all cables from the AppleTV and have restarted several times.
I have tried migrating to WiFi connection over Ethernet (no change)
My iPhone X running iOS 13.3.1 exhibits the same symptoms (except of course it is WiFi only) - When I launch the TV app on iphone I can view and play purchased content and can view home library content but cannot play any of it (same error message).
I am pretty confident that the issue is not networking hardware or permissions but software related. The issue appeared within the last couple weeks.
Not sure if any of the above 'paints a picture' for you but would love any other troubleshooting steps you might consider.
Appreciatively,
LyndyB