are you talking about the skirt board? that's the piece of 1-by that runs the length of the staircase that the risers butt/miter against, and is installed as part of the finishing/trim job. stringers are the framed 2-bys (usually 3) that literally "make" the staircase, and they are what the risers and treads (and skirt boards on exposed applications) ATTACH to.
i think youre talking about the skirt board.
personally, I think it might be tricky to run a simple straight skirtboard on an exposed side of a staircase and have it look good with the rail and everything. However, it would look nice if you were able to box out the skirtboard to give it a few inches of width for something the balusters can attach to, but you'd have to do some neat trimwork to integrate the extra width of the box to the wall below it. but, it can certainly be done!
Have you considered hybrid treads? theyre called "hybrid" because they are made out of hardwood AND regular, paint grade wood. Theyre used on staircases that are made to look like they have oak treads with a carpet runner. the hybrid treads have either just one end with the hardwood that will be exposed, or both ends in applications where both sides of the staircase are exposed. Hybrid treads are less expensive than full hardwood treads, and this way you could keep your current skirtboard AND add carpet.
my .02
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