Thank you @TaiMelcher for exploring New Quizzes and for sharing your experience. A New Quiz can be exported, and the export can then be imported into a Canvas course and converted to a Classic Quiz if the "import as New Quizzes" is not selected:
Import as New Quizzes is not selected.
I did a test with a New Quiz with one FIB question and it works, but the question reference changed to a string of characters:
New Quiz:
New Quiz FIB question
Re-imported as Classic Quiz:
Imported FIB question in Classic Quiz
Note that from a student's perspective the question works; it's from our perspective that the string of characters appears. I did not explore other question types.
I’m going to mark my response as a solution, but there can be more than one. I hope our colleagues here in the Community can share other options.