What wood color is suitable for the floor with a beech staircase?

That simply becomes difficult if you don't want any contrast (gray, white, very dark) and absolutely want to stick to a wood look. Because with the reddish beech, no other wood really matches well.

I would rather focus again on the staircase and invest money there, instead of making various compromises to somehow make the unsuitable staircase fit with the rest.

Very clear and strong colors look good with such a beech (light green, red, blue...), which is why we used core beech for the furniture in the children's room. The wood looks nice, warm, and cozy. But in my opinion, beech simply does not fit a modern living style. And a staircase is not something you can just replace easily. Now you're buying one anyway; the surcharge for a different variant will be less than completely replacing it later.

For comparison: We have two beech staircases. Replacing one with a white wooden stringer staircase with oak steps (the other would have been completely white) would have cost €17,000 at the beginning of 2022. Better to pattern-match and choose what really fits right away. We have now painted both our staircases white ourselves because the replacement cost was too high for us.