I cannot see my rentals on my smart tv

If your smart TV has the Apple TV app (not the Apple TV+ app*), then you would find rented items in the Library tab, in the Rentals sidebar, during the rental period. After the rental period, this listing would disappear again.

Rent movies from the Apple TV app - Apple Support

* The related article explains the difference between the Apple TV app and the Apple TV+ app. The latter would be available on select models where the former wouldn’t be supported.

Your app is unfortunately the Apple TV+ app, which is limited to just Apple TV+ content.

The Library tab is available in the full Apple TV app, which includes Apple TV+, Store content, and third party channels. That app would be available on different models smart TV and streaming devices.

List of Apple TV app-supported smart TVs and third party streaming devices - Apple