Darker is BETTER. Like in sewing seams on a garment, it is better to do DARKER with the tread than it is lighter. Jacobean stain has been used successfully to remove the red from red oak. If don't properly, the Jacobean stain (like water popped treads) will come VERY close to the tone/colour or the DARKER stripe in the feature wall. To keep the country theme, you can have the RISERS painted WHITE. The WHITE will help the Jacobean 'pull off' this change. A small bit of separation from the floor colour, the tread colour and the feature wall colour is all that is needed to FOOL the eye into thinking everything is tone on tone on tone. Water pop + Jacobean stain + ONE COAT of finish. Remember: the FINISH is what will add more yellow/orange (oil based or oil modified poly) or NOT (water based poly....like Bona Mega One or Traffic or Traffic HD). Find out the FINISH for the treads. You will be picking blind if you do not have a coat of finish over the stain.

Agreed with the above. For fairly priced change. I’d paint the risers and balusters white. If you want to up the game a drop you can add a runner And for the wrought iron- you will have to do a lot of it. Also at the top of the stairs by the railing in the hallway. So it’s a big project. It is expensive. Not so necessary. But it would look nice :)

Does your floor manufacturer make a matching stair cladding? If you do that, you don’t need to have oak treads. Another option is to do a beautiful contrasting color of stain on the stairs. You could pull the colors together with handrail or baluster or post colors. Adding a runner on the stairs, or painting the risers would give you more options to tie in the floor colors. If you really want to match wood, dark stain is worth a try, but lighter colors are really tough. I matched a red oak transition strip to a dark chocolate brown stone tile once. I had to keep mixing and adding stain. It worked only because it was a dark brown color. It’s hard to neutralize red and yellow tones.

I had a new oak staircase installed a few years ago. Our wonderful carpenter, who specializes in staircases, took a long time getting the right stain to match our porcelain tile wood look flooring. I am quite happy how it turned out. I think it would be more difficult to match a gray wood stain to your gray flooring.