Everybody else probably already knows this, but I have to learn things the hard way
When you create a new Moodle instance and restore a course into it, do not restore until after you have installed the Hotpot module into Moodle. If you do, the Hotpot quizzes will not appear in your course. They will not even appear in the list of course contents during the restore process.
That means, if you restore a course and the Hotpots don't restore, verify that you installed the Hotpot module in Moodle before restoring.