So what's the alternative? Small bits of rug clamped in place on each stair. These can easily be removed and washed, meeting the hygiene factor. And they can easily be replaced (to deal with staining). To ensure that sufficient grip and absence of movement on the stairs (essential to avoid injury) strong clamps are needed, but in addition to this the rug would sit on top our a rubber matting that would stand proud of the clamp.
It should be possible to design such a clamp to be both aesthetically pleasing and hold the rugs in place as tight (if not tighter) than current stair/carpet configurations.