If you paint all the wood on the staircase the same color as the walls, the staircase will have the optical illusion of seeming less prominent and noticeable. Pros for leaving it as is for now: less cost, gives you time to see how you like things in your new house. Cons for leaving it as is - it's not your own decor, expensive to change decor - what is your budget and timeline for changes? Have an overall plan before you do anything.

I’m think the first step is to rip / replace the runner with something else, and paint the walls / ceiling. Will swap chandelier when we find one we love. I hate to touch the wood until we’re sure.

Agree with changing the carpet but leaving the railings -- white paint would be inappropriate on this style of midcentury (1970s) railing. Anyway, all the trim and doors in the house are also wood tone, so you're going to have to make some broader decisions before you start painting any of it...

That's a balustrade you put on a cheap deck, not the interior of a nice home.

What did you end of doing to update the "look" of this staircase? Would love a photo. I've got the same staircase.