AUC-ROC for Multi-Label Classification
I am not sure if I understand your thinking, but my understanding is following - maybe it can help you see it from another perspective.

The multi-label classification problem with n possible classes can be seen as n binary classifiers. If so, we can simply calculate AUC ROC for each binary classifier and average it. This is a bit tricky - there are different ways of averaging, especially:


  'macro': Calculate metrics for each label, and find their unweighted
  mean. This does not take label imbalance into account.
  
  'weighted': Calculate metrics for each label, and find their average,
  weighted by support (the number of true instances for each label).


as explained here.

There is another option (which may be the closest to how you see it):


  'micro':
  Calculate metrics globally by considering each element of the label indicator matrix as a label.


which is implemented here.

There is also the fourth option, which I don't really understand:


  'samples':
  Calculate metrics for each instance, and find their average.


I have been using myself weighted averaging.