Things could not look better for you, that is a well built set of housed stairs. Since they are a housed stair you can easily replace the treads & risers. You can do this yourself or find a person who specializes in stairs or have your contractor educated himself about housed stairs.
How old is the house? I'm guessing pre-WWII. If it is most likely the glue used was hide glue, that will work in your favor. Hide glue can be reactivated centuries after it has been applied. Also hide glue will bond to itself so you wont have to remove all the old glue from the skirt when installing new wedges.
Once you have all your new treads and risers take a sawsall and cut out the old treads and risers on each side leaving about 3" sticking out of the skirtboard. Take a hammer and gently hit the face of the tread to see how solid it is. It may just slide out. Sometimes the wedges are glued to the skirt only. Rent a steamer for removing wall paper and use it on the wedge to reactivate the glue and slide out the wedge. Check the nosing profile of the new treads to see if they match the old ones. If so it could be a one day installation.