The steps down to the cellar in our last house were edged with a pretty solid slice of wood that had been set in, so that height and width weren't affected.
I don't know when this was done - perhaps it was how the original steps were designed. But it worked, and wasn't a bit unsightly.
Mind you, the steps seem to have been built from bricks, but would it be possible for a mason to carve out a recess on each of your steps to accommodate some wood? If you used oak it should outlast you with no problem, but either way it would be easy enough to replace should the need arise.
I don't know when this was done - perhaps it was how the original steps were designed. But it worked, and wasn't a bit unsightly.
Mind you, the steps seem to have been built from bricks, but would it be possible for a mason to carve out a recess on each of your steps to accommodate some wood? If you used oak it should outlast you with no problem, but either way it would be easy enough to replace should the need arise.