My (very basic) understanding of a pihole is that it calls out to an upstream DNS provider (such as the one you’d be using without a pihole) and caches everything it gets back, meaning that it’s only making new requests when you’re querying a domain it hasn’t queried before. I can’t think of any reason a game would need to constantly be accessing different domains (except maybe for some kind of server browser?)
That’s what I meant by “never connect it to the internet and plug in a more trusted device”, whether that’s a Chromecast or your PC you can always plug in something else you trust more than the TV. Obviously finding something you can trust that does everything you want is another story.