Game mode comes on for a minute then goes off on stadia
Your TV is what is technically providing game mode, though Stadia can set it automatically for TVs that are compatible with that feature. In short, I do not think it is necessarily a Stadia problem, but rather some other configuration issue.
I would try a few things:
A few things could be causing problems, and a big one for many people is that TVs that support HDR and enter HDR mode sometimes have separate settings for HDR. When it switches over, it may drop game mode. My TV does this, and thus for my Chromecast Ultra input, I needed to enable game mode for both HDR and non-HDR.