Out of curiosity I did try Customize toolbar… and was unable to re-build the search box by adjusting the position, or by removing and re-inserting it in the toolbar. It stays borked.
But what is super interesting is that once the problem has occurred, and the search bar in the toolbar has collapsed and disappeared, it also disappears from the Customize popup. That is, the draggable icon for search becomes just the magnifier dropdown, not the associated text field. The only time I ever see the pair of dropdown + text field in the Customize popup is if Search is not present in the toolbar when I first open the app. Then I can drag the dropdown and text field pair over to the toolbar, and it will work.
But only until I change views.