I'd carpet the stairs, largely for noise reasons but also because re-finishing them is a finicky job. Covering them with anything else, whether it's tiles or veneer, may not give the quality of finish you're looking for and could be difficult to match to your other finishes. The only carpet we have in our entire house is on the stairs, for pretty much those reasons! I think it's 100% wool, but know it's rated as hard-wearing - that may be as relevant as the actual fibre mix.
Another possible solution, if you really want to be carpet free, is to clean, sand and paint them. Friends of ours painted their staircase in a just off-white hardwearing wood gloss, which did look well and didn't clash with the other floor coverings. Or you could even get a combination for a classic look: painted stairs with a (well secured) stair runner.