Yes, cheetahs still live in the desert. In 2002, Mel and Fiona Sunquist in their book Wild Cats of the World wrote: “Cheetahs mainly hunt by day, but the hours vary slightly from area to area. In the Sahara, where daytime temperatures can reach 43ºC cheetahs do most of their hunting at night and in the relatively cool hours after daybreak”. The source that they refer to for the information, which confirms that cheetahs live in the Sahara desert is: A. Dragesco-Joffé. 1993. La vie sauvage au Sahara. Lausanne. Switzerland: Delachaux et Niestlé.
As the cheetah relies on speed to catch prey they need an open habitat which is normally grassy plains, open woodlands and semidesert. I suspect that the quote referred to above is a reference to parts of the Sahara which is better described as semidesert which is ideal in terms of the openness of the landscape.
In 2002, the Sunquists also reported that a few cheetahs still survived among the “sandy plains and rocky hills of the Sahara”. In that harshly hot climate they spend most of the day in rocky places and beneath bushes for protection.
In 2021 the IUCN Red List reports that the cheetah is found in the Saharan desert and we have to include the small population in Iran living in semi-desert in that country. The Red List information is good at 2015. They are badly out-of-date in my view. The distribution of all the wild cats is shrinking fast and therefore the IUCN Red List, being the principle source of information of this kind, have an obligation to update the information more frequently.