Put Fitness+ workouts on my TV
Apple Fitness+ can't be used with the standalone Apple TV app.
According to Apple:

Apple Fitness+ requires Apple Watch Series 3 or later with watchOS 7.2
or later and one of the following Apple devices: iPhone 6s or later with
iOS 14.3 or later, iPad with iPadOS 14.3 or later, or Apple TV with
tvOS 14.3 or later.

Just having the app on a TV will not work.  As you've seen, Fitness+ doesn't airplay well when run from an iPad or iPhone.  I hope this is just a bug that they'll fix later.