>>To me, this seems like it cannot be a decision they made.... surely this is a bug or an oversight.
This is not a bug. Many people seem to have the wrong understanding of Apple Fitness Plus. It's not an iPhone app that plays on your phone and gets mirrored to the TV (which is why AirPlay does not work). It's an app that runs on your iPhone or iPad or Apple TV.
I have 2 Roku's and an apple TV and I confirmed this. If you launch the Fitness app on a phone, and beam it to Apple TV - you will see the Fitness+ app on the AppleTV launch and start running.
You don't even need a iPhone or iPad to use Fitness+. If you want to use it on a TV it needs to be running the Apple TV app.
The specs say it needs Apple TV HD or TV 4K: https://support.apple.com/en-us/HT211924
The product page says "Apple Fitness+ makes it quick and easy for everyone to get the best workout on iPhone, iPad, or Apple TV. ". https://www.apple.com/apple-fitness-plus/?cid=wwa-us-kwgo-fitness-slid-
Tom's Guide says the same thing "since Apple Fitness Plus is a video-based program, you'll want a screen for watching classes. An iPhone will suffice, but you might be able to follow along better if you use an iPad or Apple TV. ": https://www.tomsguide.com/news/apple-fitness-plus-how-to-set-it-up-and-start-your-free-trial-right-now#:~:text=And%20since%20Apple%20Fitness%20Plus,an%20iPad%20or%20Apple%20TV.
Apple probably did it this way to give people an incentive to BUY an Apple TV. Apple needs things like Fitness+ to differentiate it from Roku, Amazon and all the TV's with built-in Roku or Google.
It could've been done another way, but Apple went this route -- a special fitness app on each device. They probably hope that users will want Fitness+ enough to add an AppleTV on top of their existing whatever-box.