You can get decent pre-finished hardwood for around $4 a square foot plus installation, and if this is upstairs or on a crawl space with a wooden subfloor, installation could be DIY pretty easily. Naildown isn't that hard. That's not too expensive in the flooring world, and there aren't that many choices that are less expensive that would be asthma and budget friendly. If you are downstairs on a slab foundation, you could stain it and finish it. You will need some products with VOC's though, and your son may not deal with that so well depending on the severity of his issues. There's vinyl, either luxury vinyl tile or some type of sheet vinyl. Those can also emit VOC's when new, so you'd need to research any low VOC options as well as assess their effect on your son also. There's cork, but it's more expensive than hardwood. There's tile, and you can even get tile that looks like hardwood if you wanted, but that's going to be the price of wood also, and leave you using some rugs in the winter because it's cold. Unless you install underfloor heating cables, which is WONDERFUL, but now you're well above the price of wood again, even DIYing the install. Really, wood would probably end up being the cheapest and safest choice, especially if you don't have any DIY ability.

Lots of negative waves here! :) We did a lot of thinking and measuring before we jumped in on the project. The above you issues you have all mentioned were big concerns of mine as well, but a lot of details fell in to place. I ended up ordering oak stair treads that were made to order. 1/2 thick on the main tread, return on front and both sides at 1.5" tall (to wrap existing tread), with a 1.5" return extending past the tread. They look very nice and appear to be constructed well. Prior to install, they will get stained and sealed when we get our hardwood floors refinished in a couple of weeks. Regarding liability and code, all of that has been taken in to account and they will actually be better than the original build. On the staircase, the first step from the top, just happened to have a higher rise by 1/2" than all of the other stairs, and the bottom step happened to have 1/2" less than all of the other treads, so after the retrofit they be more in code and consistent throughout the staircase than they were before the retrofit. After removing and numbering all of the spindles, we were able to take out the threaded shaft in the bottoms, cut 1/2" shorter and reinstall a wooden dowel back in its place which will fit in to the drilled holes on the tread. We made up a plumb bob with a dowel rod that fits in to the hole on the underside of the railing. When plugged in, it will give us area to drill the hole for the spindle. There is enough play on the top end of the spindle in to the railing that we will be able to raise them enough to install the tread, lower the spindle base dowel in to the hole, and then secure with a brad nailer. So far so good. Our floor refinishing got put off for a couple of weeks as prior to doing the floors, we had a contractor remove all of our popcorn ceilings in our vaulted living room, kitchen, dining room, mud room, and hallway. That should be removed and painted by within the next two weeks, and the floors are scheduled for Oct 18th. In the interim, I have dry fit all of the treads and numbered them for the steps. Many of them needed some light sanding on the inside of the side returns to properly fit over the exterior trim piece we installed on the sides of the staircase. We will be now be dry fitting, plumb bob, and drill the holes for the spindles so that once the steps are stained, we can do the final install.

Real wood or engineered. Pre-finished is a good way to go. Clear is a higher grade of hardwood that has less variation in color and more uniform with less knotholes, so not as rustic and busy looking and less of a statement. Character grade has lots of variation and knotholes. It depends on your taste. I tried to find pics of what I suggested and couldn't find the right combo, but this pic shows the lighter flooring used on the risers to tie it in. This shows an interesting riser treatment. The tile could include both wood colors. This shows darker stairs with a contrasting floor that contains the stair color and other lighter colors.

I first came across an article on using Duraseal in Country White on red oak floors to make them lighter. Then I looked up that product if it was applied to maple and white oak. Since you don't seem to want to touch your white oak kitchen floor, I think this lighter finish might be better than a dark stain and you can perhaps match it to your white oak floors later on. However I would recommend you do some testing in inconspicuous places to see the result on your floors prior to starting your project. As to your staircase, I would refinish the steps like the floors, paint the risers and spindles white and the hand rail black would work with the floor colours unified. I see you have wood trim and doors in your house and don't know what your plans are about that. Perhaps painting the baseboards and door frames in white and leaving only the doors in wood would be something to think about... See photos below: