Hi Holimoch (Or Paul, whichever you prefer),
Thanks for reaching out. This is something that isn't quite in Canvas yet as far as "pausing" tests go. Probably the closest option we have, in the scenario you mentioned, would be to "manually submit exam" via the moderate tab of new quizzes. This would make it so that the student's test is submitted at that point in time, but can be reopened at a later point in time for them to work on it later, at the teacher's discretion. This is more straightforward and less work for the teacher in the moment, though they can always intervene in the real world.
As for your suggestion, I don't believe it would "kick" the student out, though other issues may arise, such as lost submissions. I wouldn't imagine it's common to unassign students while New quiz attempts are in progress.
As for a proper pause button, that would be something that is not yet implemented, and you would want to submit that as an Idea on the community.