Since Chromecast Ultra is specified for 4K, 1080p with 60 Hz should actually work. I tried the channel and there is no judder in the normal videos. However, I don't have access to the special games. It would be worth a test for CHF 99 and the enthusiasm for games, I think.

Normally, the chromecast 2 should already be able to do this.
Of course, it may be due to something else. E.g. too slow internet connection or not enough bandwidth via wifi (whereas with the chromecast ultra you have an ethernet connection).

I can only agree with the previous speakers.
Cordially M. Z'Rotz