First, let me say that this will improve in two ways in 13.1: support of Setting and a dedicated form notebook for QuestionObject content.

That said, if you want to use the QuestionObject interface in 13.0 but have programmatic access to the answer or the assessment you can use one of these tricks:

Store the student answer in a symbol using a comparator function QuestionObject["How many career home runs did Hank Aaron hit?", AssessmentFunction[{714 -> False, 755 -> True, 868 -> False}, (MatchQ[$studentanswer = #1, #2] &)]]

Store the student answer and assessment in symbols using a wrapper around the assessment function. Note the need to specify the interface since it is not automatically inferred from the custom assessment. QuestionObject[ QuestionInterface["MultipleChoice", <|"Choices" -> {714, 755, 868}, "Prompt" -> "How many career home runs did Hank Aaron hit?"|>], ($result = AssessmentFunction[{714 -> False, 755 -> True, 868 -> False}][$answer = #])["AnswerCorrect"] &]

It also sounds like you want to you QuestionObject and AssessmentFunction together, but get access to the result programmatically. The documentation examples of nesting QuestionObject["label", AssessmentFunction[...]] seem to be created a self-contained Q&A interface, where the student simply gets visual feedback (green checkmark or red x). I also don't know how to hook into this correctly for sure.

But I think if you use QuestionObject['label", f] and tell QuestionObject your input UI type, you might be able to write a function f that uses AssessmentFunction to do the job.