The better alternative would be for the marker to be presented at login with only those assignments (and quizzes) which need to be marked by him. At the moment, the marker is presented with a cascade of menus, courses, course work and students, most of which have nothing to do with the marking task. These barriers to efficient working are a major downside to using Moodle.
I spend what seems like hours each week (probably only 1/2 hour - but the frustration!) checking the menus for work that has been submitted - through lists of assignments and quizzes that have already been marked or which have not yet been submitted. For the tutor, it is actually quicker and more efficient to use paper and the postal system, but worse for the student.
Moodle has not been designed with work flow of the tutor in mind and it shows.