I just zoomed into your stair photo and it doesn't look like there is a nose attached just kinda rounded, if you cover with a thin piece of luan or plywood you could just fill the front with putty or paintable caulk and be done with it. Or you could trim nail on a thin strip of lattice board which can even over hang the bottom so it wouldn't need cutting.
The 1/4 - 1/2" height you would add to each riser should be no problem except for the first bottom step which will be slightly higher by the added board above it - only a bull nosed inspector would cite you for that, but since the standard rise is almost global you may find that even a 1/2" difference will tip your toe, the brain is funny that way.