There are already a number of apps that do display Tweets on tvOS, so you might want to take a look to see how they are doing things. Digits will not help here, as that is not a Twitter API login, it is a separate identity mechanism. From memory, some of the apps are using a separate PIN-based flow or an external auth mechanism on e.g. a laptop or phone. You don’t necessarily need user auth anyway - app-only auth would still give you the ability to search for Tweets, for example, so there’s no actual need for login there.
I suggested Swifter and STTwitter on the basis that they are Swift and Objective-C libraries, which I understood should work on tvOS. I don’t know what alternatives there are out there at the moment.