Well, the answer is, of course, simple-stupid...
When I configured the project, I chose "Qt Quick 2.0" (or maybe 2.1), which was the default. Little did I know that there was another option "Qt Quick Controls 1.0". I found selecting that added <code>import QtQuick.Controls 1.0</code> into the QML file. I added that line, and now, lo and behold the controls are visible!
Is this documented anywhere? If not, it should be :)