You've got at least two threads started here, so it's going to be difficult to keep them both in synch. But let me try to answer some of your questions here.

I see you have four options and the pros and cons of each.

1. Strip and refinish. If you carefully mask off the parts you want to paint white and use a semi-paste or gel stripper, you can get the old finish off. Sanding is a very poor way to remove the old finish unless you are taking off a lot of wood (think sanding a hardwood floor and removing a fair amount of wood.) Sanding will not uniformly remove the old finish and cause problems down the road. You are also likely to sand out details on the railings. Sanding around the base of the spindles (on the treads) is going to be time-consuming, too. Once you have done that, you want to stay away from stains that contain a pigment as those will accentuate the open pore structure of the oak. Think of staying away from the stuff in the yellow cans. A dye will minimize the contrast in the oak, but it's still oak.

2. One way to add color after a finish is applied is with a glaze. This will probably not work for two reasons. You are pushing the glaze a lot to do that much of a color shift. And it's likely to accentuate the pores, i.e., contrast. You need to top coat a glaze with another coat of finish. Put this arrow back in the quiver.

3. The other way to add color after a finish is applied is with a toner. This is a finish with color in it. The color can either be dye or pigment. But again, you're pushing the limits of color shift by using a toner. Adding enough to get that dark is likely to make a muddy appearance. Do not under any circumstances think that Polyshades is going to do the job. While technically a toner, it's a bear to get on evenly and without streaking. And it's not recommended for floor surfaces. Toners almost need to be sprayed on to go on evenly. Very light sprays, sneaking up on the color because it goes from not quite to too much all at once. While an option, it may not work for you because of your ability to spray and the degree of color change.

4. Pull the railing off and replace with another wood with less grain contrast. By the time you go through stripper, drying, sanding, and masking, you might be cost effective to remove and replace with new stock.

Some combination of the above, such as stripping the treads and replacing the rails and toning the trim.

But remember the first rule of finishing : try out your intended finish schedules on some scrap or obscure ares before jumping head first into this, spending a lot of time and money, then having to recover from a big screw-up.

Here is a link that might be useful: Other Thread