Okay, I think I've figured out what the problem was. Put simply, it was that I didn't have Sailing. As soon as I built a road to one of my neighbors, a trade route to them popped right up on the city screen.
The descriptions of what the various water route enabling techs do is thus somewhat misleading. What Sailing, for example, actually does is enable trade via rivers *outside of your cultural borders*. Water routes *inside* your borders seem to always be active, regardless of tech. This also implies that the trade-connected icon on the scoreboard (and the ability to trade resources) only requires one of the parties to have the necessary tech, and that while they may be able to send trade routes to you, you may not be able to reciprocate.