I tried using the Chromecast on a standard BenQ Monitor in my local wifi. I can start Stadia, connect with the controller but when I try to start Destiny 2 it just gets stuck at "Checking your Connection". After 10 Minutes I tried to stop the process but I was unable to do anything. I was just told that the game is currently starting.
Had to disconnect the Chromecast to stop it. 2 additional tries came up with no different result. Could it be a Firewall issue? If so what are the relevant ports?
Do you both have the issues only with Destiny 2 or also other games?
I believe it's worth though to contact the Stadia support to get some troubleshooting steps from them as they can help you figuring the issue out.
You could also try out normal steps like router reset, run the Stadia speedtest, and try again with disabled firewall (if it is the firewall, you could isolate that issue with this).