Search button next to hamburger button on mobile?

Currently the top of every page looks like this on mobile:

This hides the search functionality within the menu. It looks to me like there is room to add a second button with a magnifying glass to better expose the search box.

This is independent of the landing page changes in #9158. Opening a separate issue to avoid cluttering that discussion.

Find the mobile header/navigation template and its corresponding styles where the hamburger menu button is defined, then trace how the search control is currently exposed in the mobile menu. Review the relevant layout and responsive CSS/JS entry points to add a second icon button for search beside the menu icon. There are no tests or file names in the issue; verify completion by checking the page in a mobile-sized viewport and confirming search is directly visible and usable next to the hamburger button.

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