No, changing (notching) the “skirting” into “stringers” will not affect the stability of the stairs, provided that 1) the new stringer against the wall is nailed to the studs in the wall, 2) the stringer at the “open” side of the stairs is supported by a wall, as shown in your “Goal” picture, and 3) riser boards are at least 1x and installed as shown.
The load of a stair is carried by the risers spanning from side to side of the stair. It is not carried by the balance of the notched stair stringers. If you look closely at a notched 2x12 stringer, there is only a 2x6 left. That 2x6 will not support much of a load when it spans 12’ or so...from top of stairs to bottom of stairs.
However, a 1x6 (or so) riser can easily span 3’ - 4’ and support a live load of 40 psf as required by code.
Often an additional stringer is installed at the center of the stair. This stringer is not to help the other stringers span from top to bottom of stairs. Rather, it’s to help keep the riser “square”, plumb and straight and to help the tread from bowing down along the back edge.