So, turns out my LG OLED TV can now download the Stadia app and play games on the TV directly, cunning.
Tried it over the weekend, it was unplayable, just juddering sound & image, so was not impressed.
Worked out how to drop the quality, as I thought 1080p might be better, it wasn't and neither was 720p, all the same juddering mess! 🤦♂️
So Googled it, I am well above the recommended connection speed, so in theory it should play 4K games just fine, I Googled some more and lots of people were reporting that WIFI is generally useless, but hard wiring works a lot better.
Found a network cable, plugged the TV into the router and bingo, flawless 4k HDR gaming with no additional hardware, noice!
Means I can stick the Chromecast on the old TV upstairs and use the app built into the TV downstairs, although I may as well use the XBox downstairs TBH as there's nothing exclusive on the Stadia.