While I'm very privacy conscious, I don't really see the benefit to hiding my region in the DNS request. Because the very next step after the DNS is my browser making a request to their webserver, at which time they will have my actual complete IP anyway.
On the other hand, it's possible this doesn't matter. The client might not encrypt the host it's trying to visit. Nation states can correlate packet timing. So if someone really wants to know, they'll probably figure it out. (This is always a risk with things like Tor. If the government is monitoring your connection and some target website's connection, and you are sending a lot of packets at the same time they're receiving a lot of packets, you can guess who is talking to who.)