Most Plasma widgets and apps that have search fields show them by default. Two exceptions are Kickoff and Widget Explorer (for background, see D15011#315779).
Kickoff is being addressed in D15011. That would leave Widget Explorer as the only case in Plasma (that I could find) where a search field invisibly has keyboard focus and appears when you start typing, but is not actually visible until then. This patch changes that by making the search field always visible, matching everything else in Plasma.
Thanks to @sharvey for doing most of the work already to move the Search Field to its own row in the layout. This patch simply makes it always visible and removes the Search button that shows and hides it.