Except, what if you posted a page of compromised passwords or other data? Then someone might go to that page and search for their own secret information, to see if it had been leaked. They wouldn't even think twice about this because normally the information in a search box is secure, and known only to the browser. But if a site creates an html based search box that is hooked into ctrl-F then the user might divulge their secret information without even knowing.
The problem here is that browsers allow overiding ctrl-f and almost no users realize this is possible. It's not just a human problem, it's also a browser design problem. Browser makers have trained their users to trust that pressing ctrl-f will pull up a find dialog, it's as much their fault as it is user's fault.
The trick is to convince someone to try finding something sensitive on the page. This might be done by showing a long list of 'leaked' passwords or social-security-numbers.
Even without explicit prompting, some people might Ctrl-F search for their own, with a false confidence that a local search isn't shared with the page.
If you're having this problem, I hear Common Sense 2013 is a great browser security package.