Your playlists are custom to your Google account. When you add content to your playlist, then it would be available on any of your devices where you are signed in to YouTube with your Google account.
Verify that you are signed in to the app. Find your user icon in the top left on Apple TV. Notice the vertical menu below that. Scroll down to Library. In your Library, displayed as a fold-out to the right, your synced playlists are listed, below a few other library categories.
If the app persists to not load content what it should load, then try force-quitting the app, then relaunch.