The Sahara is on a regular-ish cycle going from green to desert about every 30k years. It's had multiple green eras. The last one is called the African humid period
The Sahara's sand is actually full of a lot of nutrients. In fact the massive Saharan sandstorms that traverse the Atlantic Ocean are a significant source of fertilizer for the Amazon rainforest
All the elements are there. You just need the right weather and plant life to get things going.
Of course we can't ignore the human element. Archeological evidence suggests humans played a massive role in transitioning the Amazon Rainforest from the grassland it once was to the jungle it now is. Reforestation projects, Great Green Wall, half-moon berm projects, etc are all way way more effective than they were just a few years ago. We're learning a lot about how much humans can do to revitalize soils
One of these is a cyclical pattern of tens of thousands of years that caused warming and cooling somewhat less that a degree. For about six thousand years earth was half a degree warmer and weather patterns in the Sahara were different. Then there was some cooling until modern CO2 issues.
In the early twentieth century there were some very real concerns about long term climate cooling as climate science was being developed but instead greenhouse gases have blasted us very far and very fast in the other direction.