SOLVED [ALTHOUGH NOT IN A WAY THAT MOST USERS WOULD LIKE, AS APPLE LIMITS AUTOMATIC SYNC; SEE BELOW]
According to Apple Customer Support, the TV app only syncs purchased Movies and TV shows between devices. Other items must be manually added.
To sync Home Videos, for example, you must:
- Connect your iPhone or iPad to your Mac
- Open Finder on Mac
- In Finder, select your iPhone or iPad from the Locations in the lower left-hand navigation pane
- Select Movies in the top row
- Select Sync movies onto [devicename]
- Select the items on your Mac you would like to sync to your iPhone or iPad (or select all, or select Playlists)
- (Note: you cannot select items on your iPhone or iPad to sync to your Mac; this is a one-way sync)
- Click Sync or Apply
Unfortunately, Apple requires this clumsy manual sync, rather than adding a control under iOS iCloud Settings or TV app settings, and under Mac System Preferences or TV app preferences. And it is unidirectional rather than bidirectional, so there is no way to sync anything you add to your iOS device to your Mac.
Related: if you use iCloud Music Library, you cannot sync songs from your Mac to your iOS device (or vice-versa). If you want to download songs to your iOS device, you must open the Music app on that device, then select Library in the bottom nav, and then download each song individually. Argh. Another very poor product design choice. And easy to solve: They could allow selective sync from the Mac using a list with checkboxes just like in Movies as described above. And on the iOS device, they could create a mode allowing you to select multiple songs and then do multiple downloads all at once.
Bottom line: Yes, you can muscle your way through this. But, as very few people actually do a manual sync, instead relying on iCloud, this should all really be in the iCloud settings or app-specific settings.