I suppose what the author really means is the end of the dominance of the American upper class, since they are still there, just lower key. Nonetheless, Philips makes the claim that the transformation of American society into a meritocratic one was caused by the return of the officers from WWII. Unlike in other countries, you could be an officer in the US on the basis of ability, and when these men returned, they could go to college with the GI Bill, or go to work in the post-war boom.
I suspect that class is one of the universals of the human condition, and that it will eventually restore itself by means of natural feedback mechanisms. 50 years after the last Social Register was published, there is a mass upper-middle class defined by college attendance and professional work that was enabled by the events Philips mentions. But there is nonetheless a congealing caste of families that dominate the upper echelons of both business and government that is defined by attendance at elite universities, and often by subsequent residence in a few great cities.
The reasons for this are largely hereditary, in the fullest sense of the word. Both genetics and upbringing are required to give the ability and the willingness to spend one's twenties [and maybe thirties] in the credentialing institutions that grant access to the mass upper-middle class. The process that extended this opportunity down the socioeconomic ladder has ground to a halt. This process is self-limiting for several reasons. Economically, all the slots for professional work have been filled. Furthermore, you still need people to clean stuff, make stuff, transport stuff, and fix stuff. AI and robotics turned out to be a lot more difficult than originally imagined, so this will still be true for the foreseeable future. Educationally, college education is not appropriate for everyone, and so the object of a college education cannot be fulfilled if you extend college to everyone, because you will have to alter it. Finally, since we tend to marry people like ourselves [assortative mating], the ability and willingness to attend college are becoming concentrated in the children of those who have already gone to college.