Views table rows not draggable when Weight is first field

On the site I'm currently working on, my views for weighted content types are not draggable when Weight is the first field; the module is putting the drag handle into the Weight field, but it's also hiding the Weight field because it thinks it only has the dropdown inside it.

I create very similar Weight-sorting views on every site, so I'm not sure how I haven't run into this before. Maybe I've never bothered to make Weight the first field? Seems unlikely, though.

At any rate, if this is a global problem, the JavaScript should make sure it's not sticking the drag handle into the same field it's hiding.