This question has been fully answered (the expected number of moves is $\infty$), as detailed in an addendum to the question. I place this community-wiki "answer" here so I can accept it (It let me!) and so (hopefully!) prevent the MO software-bot from re-asking the question.

This question has been fully answered (the expected number of moves is $\infty$), as detailed in an addendum to the question. I place this community-wiki "answer" here so I can accept it and so prevent the MO software-bot from re-asking the question.