Does the Apple TV App need the Apple TV Box to work?
It does not need that per se. Select reasonably modern TVs support an edition of the Apple TV app among their smart apps.
Watch in the Apple TV app on these smart TVs and third party streaming devices - Apple Support
Apple TV app vs. Apple TV box - Apple Community
I have had a problem using the box where it keeps disconnecting or stopping, the programme freezes, so have resorted to Airplay from my iPhone to watch.
Sorry to hear that. Disconnecting or stopping is often associated with network issues. But if AirPlay through the Apple TV box works, then the box would have good network connectivity. Does it happen for one app in particular, or for any app?
Please review: If you can't watch or stream movies and TV shows in the Apple TV app - Apple Support
Else contact Apple Support for assistance.
I recently stayed in a property which had the AppleTV App on the Toshiba Smart TV. No AppleTV Box was evident. No problems as described above occurred.
QED. It works without Apple TV box too.