Aha! According to this tech note, you have to push in the remote's connector extra-special hard in order for it to go "click!" and start working. Pushing it in only until it's flush with the unit (as one normally does with plugs of this variety) will allow music to flow through but the buttons won't work, which is the phenomenon I was observing.
I dunno. This seems like a misfeature to me, since it's not really obvious that you need to push down on it so hard when connecting -- the included paper doucmentation doesn't mention this at all.