I had cork stairs in my house when I bought it. It was done with the same cork panels that were in the kitchen. I had it all ripped out and my stairs rebuilt because it was so noisy. Now some of that was the underlying structure, but some was because the cork was not really part of the stairs. It also did not wear well.
I’m not sure how cork could be installed and look nice, but maybe just for the steps and not the risers? I’d look into how this would be done and see if you can find some good examples of the installation.
Cork is a great product – renewable and environmentally friendly. But it needs to be done right and not in an area where there is a lot of foot traffic in patterns or where water can be spilled. I put in Marmoleum in my kitchen to replace the cork – another green product and, I think, much better choice for a kitchen.