For Chrome It behaves like you said but for some reason on mobile and tablets everything gets messed up. Part of the problem is that 50% of my users are mobile and it is breaking in mobile. Also, on IE and Edge the search does NOT show up in the menu, it shows as light blue on the side search bar and the widget as white in the footer. I tried the same with Firefox. Something is still strange.
Can you take a look? You still have access when you get a chance.