If you already subscribed to Apple TV+, then your Apple ID account and password have already been verified to be in working order. So that part is fine.
Please do not create another account for this use. That is not the solution, and it can make things very complicated in the future. The subscription service is only for the account that was used to subscribe. When two accounts have the subscription, then two accounts pay for it.
You do not need a smart TV. The Roku streaming device will take over the ‘smart’ part anyway, ignoring any built-in smart functionality. Many Roku users can successfully stream through the Apple TV app, and it can work for you too.
If you can't stream or watch movies or TV shows from the Apple TV app - Apple Support
Besides the mentioned checks in the linked support document, I would add to check for software updates to the device and the app. Generally, Apple software and services doesn’t aim to be compatible with anything but the latest versions available.
If you activated two-factor authentication, and if you sign in newly on this device, then you may be prompted for a two-factor authentication verification code. That code would be send to a trusted secondary device, e.g. a mobile phone. If that part takes too long, then the code may lose its validity, not recognizing the sign-in, and for you to try again.
