Does the X3000i not auto-detect whether you are feeding it a Game signal or a regular signal?

I have both an Xbox Series X and an Nvidia Shield hooked up to my X3000i via a Denon X3700 AV receiver, and whenever I switch between the Xbox and the Shield (which is very regularly) I have to manually change the picture from Game to my movies setting and then back from my movies setting to Game mode again. Really thought this would be an auto-detect thing like my TV does.

If I'm in Game mode with the Xbox and then switch to a HDR movie on my Shield, it will detect that it is HDR10 but it stays in Game mode and goes to Game HDR10.

Similarly if I'm watching something on the Shield in Cinema mode and then turn on my Xbox, the X3000i just stays in Cinema mode and I constantly have to change manually. Am I missing something in the settings? It's pretty annoying