The stairs should be considered part of the 'public' areas. That means they should match/compliment the downstairs area (the place with the entrance, the living/dining space, etc) where guests come in and move around. Traditionally 'upstairs' would mean private space such as bedrooms. The upstairs areas are welcome to have different flooring from downstairs. That's not a problem.

You are welcome to work with treads that are 2-3 steps darker than your downstairs flooring and then paint the risers. This is a very handsome look that will show off both the stairs and the down stairs flooring. If you are lucky, you will be able to pick a stain/finish that is complimentary to BOTH floor colours.

Of course, a stairway that has been designed for carpet has a different set up than one designed for rigid flooring. You will need to be careful with how the finishing carpenter finishes the stairs.

Skip that icky dated gray anything and match the floors and stairs to the existing wood floors. Be sure that you don’t have more than a 3/8” variance in those stairs, or you will be rebuilding the whole thing to modern codes.

The reason we're wanting to go a different & lighter color is because of the furniture we have upstairs. It's all dark wood furniture so we wanted to do a contrasting flooring.

Go lighter with the upper floors but stay in the same color family as downstairs. You will likely need rugs in the bedrooms which will also help break up the dark woods. Match the stair treads to the lower floors, with white risers.

From these comments, we're likely going to go with the same color throughout.

The lighter shade of what we have is one we really don't like much, but I totally understanding that this current color is timeless so it's a safe bet.

If it's in the budget, you could sand and refinish your first level floors to get the lighter color you like, and then match it upstairs.