Which is a more sustainable choice: hardwood mulch or rock?
You are correct in your assumptions. Mulch is better for your plants; it provides nutrients, moisture, and other benefits. Rocks are better for your home; they do not attract as many bugs and they drain water well. I would recommend a combination of the two around your house and plants.
As far as how sustainable they are, it depends: rocks have an almost infinite life cycle and can be used again long after your home is gone, making them very sustainable for green landscaping. They do require tremendous energy to mine and transport, but with a little research I am sure you can find a local rock quarry with little erosion. Mulch is also quite sustainable if it is from trees or plants that would have found fate in the landfill or been burned. Again, with a little research, I am sure you can find a local source for mulch. Ask your local wood chipper company what they do with theirs.