If you paint with a non-yellowing wood finish and/or perhaps a light coat of stain in your preferred colour you can then keep the same throughout which will make the house flow

It is a funny stair with a mixture of styles. The open riser is a more contemporary look, whereas the newel post, balusters and handrail have a very traditional look. Painting it white will not improve this mismatch of styles. If you paint the treads white then you will be regularly re-painting them as painted stair treads wear back to the timber quite quickly with foot traffic. From your photos it looks like you have a traditional, older style home, so you may want to consider replacing the stair with a new stair that has a more traditional look that doesn't look like a stair that was added in cheaply by a builder who had no idea when your first floor extension was done. However, a traditional stair will make your hallway seem more closed in without the open risers, and having the underside enclosed. The other option is to go very contemporary with steel and timber which is visually lighter and is in contrast with the rest of the entry hall. It is a really important visual element to the entry of your home and a coat of paint is not going to fix its problems. Best of luck with your considerations, Dr Retro of Dr Retro House Calls

Is the sitting room at the top of the stairs? If so you could carpet the top riser. See attached photo.

Probably would need timber staircase but have you thought of a sisal runner up the stairs as an alternative to carpet? It keeps the noise down and is very hard wearing