No. Of course not.
The privacy-specific threat of DNS is a bit more general an nonspecific: A person at IP address
X visited both Facebook and Spotify early Friday evening. It doesn't really tell you a whole heap unless you actually have Facebook's and Spotify's logs to try to do some correlation. And of course if you have that, then the DNS information is redundant and completely unnecessary.
But more importantly, the real threat here is your ISP, not Google. Your ISP traditionally runs your DNS servers, and can view, intercept, and modify your DNS queries and responses no matter whom you send them to. But your ISP also is in a position to log all of your traffic patterns; what sites you visit and when, how much data you transfer from each, and if you're not encrypting everything, then the full contents of .. everything.
And in real-world cases that actually mattered, Google has consistently taken the site of protecting the privacy of its users, while many ISPs have consistently betrayed the trust of their users.
In other words, you're taking extraordinary action to protect yourself from Google even though Google is typically trustworthy, while you're not taking any action to protect yourself from your ISP, even though your ISP is typically not trustworthy.