The Paramount+ subscription as started outside the Apple TV app, is not supported for the Apple TV app Channel Paramount+. These are different types of subscriptions with different terms and conditions, and different perks and limitations. Paramount+ also mentions this on their website.
The Paramount+ subscription as started outside the Apple TV app, will require the separate Paramount+ app to be installed. Unfortunately, that is not supported on the Apple TV (3rd generation). Installing new apps is supported on later models.
I couldn’t find a definitive answer on AirPlay for this content provider to an Apple TV (3rd generation), but do try that.