You can get Oak doors with less grain, it depends on the type of Oak. We had Oak doors in our last house, and I loved them, sorry, but I forget what kind we had,(maybe 'Red Oak'?) definitely not a lot of grain. Though I wasn't really happy with the colour the GC had us stain them. We did the work ourselves, but we were young and too easily influenced, and went with his recommendation instead of our own gut feeling. We were trying to get them to match our prefinished oak floor, and they ended up way too orangey. But we were 'house poor', and couldn't afford to redo them. Moral of the story- go with what YOU want, not necessarily what the GC suggests.

Can you show the living area next to the staircase? You have so many color elements going on--golden brown tile, reddish brown steps and banister, a patch of turquoise, what looks like navy and white rug, yellow chair. So lets look at the fixed elements: gold/brown tile and steps and banister. The runner should go with those basics with some possible small areas of color. A golden sisal with reddish brown binding could work. Or a print or tweed in golds, reddish brown, a bit of yellow. This is a jute:

The above picture looks lovely as the dark banister ties into the darker inlay in the hardwood landings. The basement has none of the above and it actually has a wood look tile floor. You will not get the same lovely result. I think it would look very odd. You would have wood look tile, stair tread stain and yet another stain. IMO It would read a jumble of finishes.

it's pine. pine is really horrible for staining. gray is going to be a no. it just doesn't look that great on pine. sand them well, 120,150, 220. wood conditioner pre-stain. Then you stain. get a similar piece of wood to try on some samples. better just to finish the wood as best as you can, and do a natural finish. pine can stain blotchy. here's a darker color http://temptingthyme.com/2014/04/16/staining-pine-stairs/ Pine just doesn't do well w/darker stains. and gray? pine is too yellow for a light gray you can try it, but be prepared