Am I going about this completely the wrong way? - should I not be using a quiz at all? - is there no concept of having completed a course?
hi
you are not going in completely wrong but somewhat wrong way

you should be using lessons and not quiz.
Then you may do all of the above.
Completing a course means different things for different users.
if you want to unenroll them from a course,preventing future access, you may unenroll manually, or leave it on students to unenroll themsaelves or ask moodle to unenroll them after a certain period. The course itself can be set to complete and expire on a certain date.
The
lesson can be set to be taken once only and thus students who are still enrolled but keep re-visiting cannot re-take the
MCQ lesson. So it depends what is it that you want to happen upon completion of course.