What is the exact model number of this smart TV?
There are two ‘editions’ of the app:
- the full Apple TV app (with Apple TV+, Apple TV channels, and iTunes Store purchasable movies and TV shows),
- and the limited Apple TV+ app (with just Apple TV+ content – no access to purchased movies).
The difference is further explained here: Get the Apple TV app or Apple TV+ app on your smart TV or streaming device - Apple Support. Your smart TV may have access to one of these, based on exact model (TV system software version, introduction year). Unfortunately, the Apple TV+ app wouldn’t have a Store tab or Library tab to find your purchased movies.

To use AirPlay, your TV or TV-connected streaming device must be AirPlay-enabled, and have that activated in TV settings.
List of AirPlay 2-enabled smart TVs and third party streaming devices - Apple
Of course the Apple TV box is also AirPlay-enabled.