I can verify it works on Home/Nest Hub. I've always said "on <Chromecast Name>" but if you have the Home/Hub "default TV" set up in the Home app it may not require it, though I've actually never tried omitting it.
I've started Destiny 2 this way and it will wake up the TV, switch to the Chromecast input, start the game and wait for you to turn on the controller.
@Trainz Have you tried something like "Hey Google, play Destiny 2" instead?
I can verify that works with the Assistant on my Pixel Slate. With Google Home, you might need to include "....on [Chromecast name]" as well.
I can verify it works on Home/Nest Hub. I've always said "on <Chromecast Name>" but if you have the Home/Hub "default TV" set up in the Home app it may not require it, though I've actually never tried omitting it.
I've started Destiny 2 this way and it will wake up the TV, switch to the Chromecast input, start the game and wait for you to turn on the controller.