Or broken DNS is so pervasive that it is interfering with their ability to offer other services. If you're interested in the privacy policy around Google DNS it's available here. The quick TLDR is:
What information does Google log when I use the Google Public DNS service?
Google Public DNS complies with Google's main privacy policy, which you can view at our Privacy Center. With Google Public DNS, we collect IP address (only temporarily) and ISP and location information (in permanent logs) for the purpose of making our service faster, better and more secure. Specifically, we use this data to conduct debugging and to analyze abuse phenomena. After 24 hours, we erase any IP information. For more information, read the Google Public DNS privacy page.
Is any of the information collected stored with my Google account?
No.
Does Google share the information it collects from the Google Public DNS service with anyone outside Google?
No, except in the limited circumstances described in Google's privacy policy, such as legal processes and enforceable governmental requests. (See also Google's Transparency Report on user data requests.)
Does Google correlate or combine information from temporary or permanent logs with any personal information that I have provided Google for other services?
No.