When I type "www.google.com" into the search string of my search engine and hit "I feel lucky" then it gives exactly the same result (and that's how I saw a few people navigate to sites).

So if we go with the path of UI simplification, we could as well get rid of the URL (what does this TLA mean for a user, anyway?) and just leave a single input-only field.

Wait, that's exactly what is there today after you click on the icon with the house on it... So, maybe just make a checkbox "hide the URL input" and "show search engine homepage on new tab" ?