Firefox

This looks promising. Some of it is half-cooked, but the developers are soliciting feedback and actually responding to it there.

The dropdown should only be visible when the search bar is focused or the new tab / blank page is open

There is work being done to implement that behaviour

Back to the post, Mozilla also poses this question...

How Does This Benefit You?

...before providing some great answers. It's good to see Mozilla still knows its target audience(s) and is still capable of communicating with them.