Timber floors with just a quality sand and polish go with almost any design style. Even any species almost works ok with anything. Unlike tiles......

Definitely depends if the ground floor is slab on ground or lightweight structure. If it is lightweight than concrete isn't an option anyway! Concrete is a hardy material, but it needs to be protected during construction. Timber integrates that warm natural feeling that can often make or break a home! What materials are being used for the remainder of the design. If there is already a lot of timber, it can be beneficial to incorporate concrete or tiles so that the timber isn't overbearing. If the remainder of the palette is simple white plasterboard, than the warm timber floor is often beneficial.

I’d go with timber or cork. Cork is great in a kitchen. Easy on your feet and joints and things are much less likely to break if you drop them. Especially as you age, tiles cause fatigue if you are standing in the kitchen for too long. Cork is now available in lots of different colours and styles including charcoals and greys that might compliment your slate. It’s not all 80s orange. Of course timber can look great too. Just make sure it’s ok for kitchen use.

At first I thought OMG what are you thinking with putting a grey kitchen in with all that beautiful wood. But I did a google search, and found the colours look good together. I would keep the floor a neutral colour and pattern. Not concrete look, not wood look, definitely not slate look. Refer to the photo below, a neutral floor with the grey kitchen, let the wood be the feature.