I would wait until you finish kitchen before deciding on the floor.

Really, the sky is the limit. Would you like a pattern? What colors domyou have around your home and in the sitting room?

Try some flooring out virtually on your photos so you can narrow down your choices.

The furniture in the house has sage and ivory / beige - neutrals, for the most part. A pattern on the tile would help to limit constant visibility of anything that gets tracked down the hall. Thank you for the pictures, @BeverlyFLADeziner - that is so helpful! Looking into software is a great idea.

No software required. The “visualizer” is a feature available on suppliers or manufacturers’ web sites.

If you don't want dirt to show, scoop up a couple tablespoons of dirt from the yard into a ziploc bag. You can also add a bit of cat hair fluff from your vacuum. Take that and a birch wood scrap to the flooring store with you. Choosing a flooring color close to the dirt will hide dirt.

Are you intending to put a temporary patch for safety on the missing tile area at the base of the stairs?

Honestly I’d say start by looking at your kitchen finishes as this floor will be the base of all your decisions. Don’t worry about matching tile to wood stairs - that’s not a thing you do. But you do want a kitchen floor that is a god base for your counters and cabinet finishes.

I see no saved images of kitchens in your ideabooks. Start there - so, keeping counters or not is your first decision. Then pick a floor that works with that decision. I guarantee it will look fine with your wood stairs.