Aha! Figured out the issue - apparently the new version of React has an "onclick" handler on the root element, thus Draggable tries to politely stay out of the way by opting out unless you set dragClickables:true on your Draggable. Or you can set your own clickableTest function, but that's more advanced. So this isn't a bug in Draggable - it's how it's supposed to handle elements that are "clickable" (outside of Draggable), and it's easy to override if you need to.
In short, just add dragClickables:true and you should be golden. I'm not sure why React is adding that onclick now to the root. Anybody know?