Some times the stair trim (skirt board) is installed attached to the stair stringer, so it may not be easy to remove. Regardless, I would keep the stair trim there and veneer it with about 1/8" or 1/4" plywood and add a solid wood cap. Do this first, before adding the new stair risers and treads. The stair skirt is not the easiest thing to veneer, so it is easier to cut the stair risers and treads tight to the skirt than vice versa.
Make a pattern from cardboard. Cut these strips a bit wider than the stair skirt board to allow for scribing against the existing stair tread and riser. Once you are happy with the fit, trim the veneer off flush with the top of the stair skirt board and finish it with a piece of oak, or whatever choice of wood, running on the top of the stair skirt, acting as a cap to cover the end grain of the veneer and top edge of the old skirt. Then fit the new stair treads and risers to the skirt. Watch the height of the stair risers. You don't want to change these too much - your stair risers should be consistent in height from floor to floor.
Dave(Ask Dave) (About Dave)
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