You are correct that questions in Classic Quizzes cannot be moved from a quiz to a question bank. A workaround is to export the quizzes. There is an option on the Content Exports page to select quiz instead of the entire course, and this creates a QTI file that can then be reimported into the course. This might duplicate the quizzes but, for your purposes, also creates question banks. On the Import Content page, make sure you select QTI for the content type and then the option to create new question bank.

Import Content dialog with QTI and new question banks selected

This creates a new question bank for each quiz in the export file, and from there you can move the questions around.