Thanks for your reply! The mystery deepens...
I tried all of those steps, with no luck. I have no extensions in Safari so that was short. When I tried troubleshooting around a different internet connection, things got really strange.
I tried watching on my iPhone instead, using Safari instead of the TV app, and got a "There was a problem loading this content" error. When I tried watching the same content on the app, with the same internet connection (not cellular data), it worked. When I closed that and tried it again on Safari, it worked this time. Meanwhile, on my Mac, leaving the black screen up to see if would load resulted in the Mac rebooting (it doesn't do this every time, usually it just sits there).
Is there any chance this is some kind of DRM test near the beginning that my iPhone passed by streaming it on the app, so that it worked afterward on Safari? I don't have the option of doing that on my Mac, as the TV app isn't available on El Capitan. I don't think it's the internet connection, since Safari on that connection will work on my iPhone. Strange stuff...