What's private about a web search, exactly? For unauthenticated sessions, Google does not use HTTPS by default. If I click through to a SERP link, my search referrer is passed to that party. They can do what they want with it, including alerting authorities. Not that authorities need alerting or warrant to capture all non-secure traffic.

I'm not convinced we know any of the reasons for the visit yet, but it's fairly disingenuous to consider ANYTHING done on an online property, that you don't own, is not public.