Should I do a dark dark blue on the treads?

No to blue.

I think a dark charcoal would work, but keep the tile white. Or could you strip them to the natural wood and do a dark stain?

Make sure your prep the stairs well, since it will get a lot of wear and tear.

Do you have a mat, rug, art nearby? What do you think of the neutrals there currently?

I would strip the stairs and go with a stain also. What area of the house is this stairway a part of? Are all the walls similar?

Safety trumps aesthetics; would bring railing up to code before worrying about colors. Handrail should extend to edge of last stair. Disclaimer: Have spent my career in trauma centers seeing and treating people killed by falls on stairs; stairs need to be as safe as possible. Building code is the minimum.

Plus, note <4" sphere requirement.