Lilian replied off list (please don't do that, I would much rather people replied in the forum) to ask what next.

The only way to fix this in your own Moodle site is to apply the techniques from http://docs.moodle.org/dev/Themes_2.0_overriding_a_renderer to change what the view_page method in mod/quiz/renderer.php does. The three lines of code

        $output .= $this->view_table($quiz, $context, $viewobj);
        $output .= $this->view_best_score($viewobj);
        $output .= $this->view_result_info($quiz, $context, $cm, $viewobj);

needs to be changed in some way. You might be able to just delete one or two of those in your overridden class, or you might need to override the methods being called.

If we wanted to fix this in the standard version of Moodle, then similar changes would have to be made in the main quiz code, and submitted as a patch.