The search box provides suggestions, but it does this by sending everything you type to your search provider. The address box doesn't, because we feel it's bad for privacy to send every web address you type to a third party.
On the other hand, if you could make this work, I'd love to see it.
Alternatively, although not intuitive, if the user enters focus with the search bar through a specific key-stroke: CTRL + L for a regular URL entry, and re-route CTRL + K to our web-bar, but with suggestions.