I'm not so sure. Primitive mining in e.g. Røros, Norway goes back to 1740. Dynamite were introduced to the mine in 1870. https://en.wikipedia.org/wiki/R%C3%B8ros
Only 40 years before the invention of the Geiger counter. If future primitive people have dynamite, it probably won't be long before the know about radiation too. This whole project seems to be being treated as far more important than it really is. It's only to protect a very small group of people in a very narrow window of technological advancement and only if civilization had somehow already been destroyed yet there were enough of us around that living in the desert seemed like a good idea.
I agree. If you missed it in the article, it's even bigger than that. The last sentence is most telling:
"This panel member therefore recommends that the markers and the structures associated with them be conceived along truly gargantuan lines. To put their size into perspective, a simple berm, say 35-m wide and 15-m high, surrounding the proposed land-withdrawal boundary, would involve excavation, transport, and placement of around 12 million cubic meters of earth. What is proposed, of course, is on a much greater scale than that. By contrast, in the construction of the Panama Canal, 72.6 million cubic meters were excavated, and the Great Pyramid occupies 2.4 million cubic meters. In short, to ensure the probability of success, the WIPP marker undertaking will have to be one of the greatest public works ventures in history."
Plus, as interesting as this project is, I can't imagine coming across a gargantuan spike field in the middle of the desert that I would not want to explore.
Mining is much, much, much older than 1740. Basic cave mining dates back to prehistory, organized tunnel mining dates back to Ancient Egypt and Greece, and the Romans used sophisticated large-scale hydraulic-mining techniques with the use of pumps to drain water from deeper mines. By the medieval period the Germans were hitting really tremendous depths - by 1700 or so many mines were hitting depths of 300+ meters.
Imho, that's still solvable easier than creating marks readable for 10000 years in the future. Simply choose a mountain that contains no valuable metals/minerals and is made of rock that's too hard for primitive tools.