Netflix wouldn’t have been allowed on iOS devices if appleTV would have been out first :)
And even so, you're not allowed to sign up or renew your membership via an iOS app. Heck, I don't think you're even allowed to link to a web browser page from the app itself. Well... you are allowed to pay from the app, but then Apple wants a 30% cut to enable that. The video streaming services have balked at paying that. Rightly so.
Worth pointing out that Netflix has even had a game-like interactive movie called “Black Mirror: Bandersnatch”.