I've been riding a Raleigh Capri for about a year. I like it, but a couple of months ago, it started popping spokes.

My commute involves a bit of rough road, including some mini-potholes and repair spots that I can't always avoid in the heat of rush hour traffic. It seems like the wheels just started breaking down from wear and tear, which brings me to my main question:

Is a road bike even appropriate for my kind of riding, that is, if I bought new rims and/or spokes, would it just break down again in another year because I should be using a heavier bike?

A well built wheel will have no problems and it sound like your spokes are not tensioned enough. I only have road bikes and I ride on many rough streets without issues. You can try getting wider tires that will be more cushiony than 23mm tires at 120 psi, but those work too