Remote Desktop (RD client, AnyDesk, AirServer, etc.) on Ipad turns into gray/black screen after turning on AirPlay

I have different applications installed on my iPad to connect to a remote desktop on a Windows server. I successfully connect and work with applications until I need to show students a Remote Desktop image via a MacBook or Apple TV.

As soon as I enable the AirPlay option, the remote desktop screen on Ipad becomes gray/black (depending on the RDP client) and I can only see the desktop image on Macbook/Apple TV and can continue to work with remote desktop on Ipad using my finger or Pencil on gray/black screen, but seeing all actions on MacBook/Apple TV.

Through experience, I found out that the problem was definitely not with the MacBook or Apple TV, since I broadcast the image on a Windows PC (via AirServer) and the result was the same. It's also not a matter of the Microsoft RD client version, since the same thing happens with AnyDesk. The problem occurred on iPad Air Gen5 (iOS 17.2). I decided to check the iPad Air 2, on which iOS cannot be updated above 15.8 and this bug is also reproduced there. But Mirroring works great on an iPad Air (Model A1474) running iOS 12.5.7.

Does anyone have a proven solution for how to broadcast a remote desktop from an iPad to an Apple TV and so that you can work on the remote desktop using Pencil too?