Apple TV (3rd generation) [A1427 or A1469] from 2012 should still work for the functions that it has, but no new software updates will come, and no new services will be added. Content providers may choose to abandon this model, as various have done over the years.
Users could never add their own choice of apps/channels to this model; that came for later models than have an App Store.
The Tablo TV app that I could find is available for iPhone and iPad, not other Apple devices at this time.
➔ A version for Apple TV (4th generation or later) is said to be “coming soon”.
If you meant a different app with the same or similar name, then please provide more details.
You can try if AirPlay is supported? It does require support from the content provider. Some content providers do not make any provisions for AirPlay support. Other content providers choose to block AirPlay. Yet other content providers may have minimum system requirements for the receiving device that exceed Apple TV (3rd generation).