These days you have no chance of getting any form of tech support or even issue acknowledgement unless you have a large follower count online.
Especially if you're using it to, say, watch YouTube, as Paul Vixie was.
Chromecast connecting to Google is not a "secret requirements that go far beyond user expectations." Expecting a cloud-connected product to continue to work when you've randomly blacklisted IPs that belong to that company is, however, an unreasonable expectation.
https://security.stackexchange.com/questions/62273/can-dns-s...
It is similar in some ways to Facebook tracking users across the internet through Facebook "Like" and "Connect" buttons across the internet, even when the user's aren't on the Facebook site and did not opt-in to having their browsing tracked by Facebook outside of Facebook.
It's a Chromecast. You literally can't make non-Google queries on it at all in the first place.
So what, exactly, is the privacy concern here? What specific flow of events does the Chromecast using 8.8.8.8 impact your privacy?
Why would ______ want to do _______ with their ______ device?
You can make a locked down device that only does a very limited subset of functions, but you really should make that known to the user before hand. "This device requires access to $X servers to function".
If you have secret requirements that go far beyond user expectations, expect that your users might get pissy about it.