Mount Kailash is sacred to four traditions at once, Hindu, Buddhist, Jain and Bon, and the circuit around its base is walked by pilgrims of all of them, clockwise in three cases and anticlockwise in the fourth. Nobody climbs it, by agreement.
Alex McKay's history of the site makes an argument worth recording carefully, because it is the kind of finding the atlas exists to hold. The identification of this particular peak with the Kailash of the texts, and the shape of the pilgrimage now practised, are considerably more recent and more constructed than the devotional literature suggests, and the modern circuit owes a good deal to renunciate traditions that established themselves in the western Himalaya in the last few centuries and to the accounts of British surveyors and travellers who described what they found in terms their readers expected. This is not a claim that the sanctity is invented. It is a claim that sacred geography has a history, that it can be dated, and that the dating is often much later than the tradition assumes.
The practical questions are modern too. Kailash lies in Tibet, so Indian and Nepali pilgrims reach it across an international border under permit, and access has repeatedly been suspended for political reasons or reopened as a diplomatic gesture. Whether a pilgrimage exists in any given year is currently a matter for two governments.