Except in this case nobody is controlling your DNS, as Chromecast doesn't let you make arbitrary DNS requests via it. So Google/Chromecast only knows what DNS

Except in this case nobody is controlling your DNS, as Chromecast doesn't let you make arbitrary DNS requests via it. So Google/Chromecast only knows what DNS lookups Chromecast makes, which changes nothing with regards to privacy or anything else. It can't watch what you're doing, it can't snoop on your web traffic, etc...