If you like privacy, why not use Safari ?
Looks like I'll be giving Safari another try.
You can actually find in page by typing a keyword in the address bar. There will be an "On this page" section at the bottom of the suggestion list, which you can tap in and jump through all matches. If I remembered correctly, this feature has been available since iOS 5 or so (which were a little bit clearer back then, since search box was separate from the address box).