Here is a work-around for those who would like this feature. Canvas now supports exporting group membership by CSV (without having to resort to using the API). Export your group membership and then re-import them as SECTIONS. To do this, go to "People" and then (with mild modifications to the exported group-membership CSV) import your students to sections with the same names as your groups. You can then assign the quiz to different sections. 

This manual group-to-section bridge is far less tedious than adding students one-by-one to each quiz. It is still not perfect.

I have taken this approach to generate a kind of "sign-up" for different assignments (similar to the sign-up options in the calendar for office hours/meetings). I turn on self-enrollment for a certain group set. Then, at a deadline I give them, I convert all of the groups in the group set to sections. Each section is associated with a different assignment option (which might only vary in due date).