The Chromecast has always been a locked-down device. I suppose it's frustrating if unsecured DNS was the one way that you could exert some control over a device

The Chromecast has always been a locked-down device. I suppose it's frustrating if unsecured DNS was the one way that you could exert some control over a device that isn't truly yours, but that's just not a battle than can be won. Your beef should be with the Chromecast, not DoH.