Okay, I did some troubleshooting. This is what I see now when I try to add a featured story after clicking on “choose a story.”

Not every browser is supported properly. In Safari, I had to type in letters to search for a story, but if I randomly clicked in and around the “start typing to find a story…” bar, sometimes the list would pop up.
In Firefox, there is a little arrow centered below the line indicating a pull down menu with the list.
And if you have a mobile device, I have no idea what’s going to happen. Just know that typing in the first letter of your desired story does work.
Mystery solved.
UPDATE: I just figured out that if you click anywhere just below the line of type, the pull down list will appear. If you click it again, it will toggle back off.