I reset my Apple TV and it didn't solve the problem.

I too am having the same problem. The Mac appears as paired on the Apple TV, but the Developer Menu does not appear. I have reset the Apple TV to no avail. A bug perhaps in Xcode 13.3?

After moving recently, and updating both my Mac and my AppleTV, I can link my Mac to the AppleTV, but the Developer menu does not appear. Using Xcode 13.2