How did Academia create and/or solve the chicken-and-egg problem of having a PhD?
The title PhD as the name of an advanced degree was first awarded in Germany at the Humboldt University of Berlin in the 19th century. From there it spread to the US, Japan and elsewhere in Europe. 

The chicken and egg problem was simple: you don't have to have a PhD to certify someone as an expert in some field of knowledge if you are already a world-leading authority in that field of knowledge. The faculty at the Humboldt University at that time were all already widely known experts, and so they were able to credential other as experts.