Some of the New Chronology is delightfully whacky. In particular I love the idea of historical events documented by independent sources result in appearing to be two or more events that cause the timeline to be expanded. One example is that Suleiman the Magnificent and Solomon were the same person, just documented from two different perspectives. And thus the Hagia Sofia and Solomon's Temple are the same building.
His criticisms of radiocarbon dating are interesting (while not completely supportive of his hypothesis). But it is an interesting rabbit hole to dig into; the relationship between radiocarbon dating and dendrochronology, and the attempt to get an accurate historical record of carbon isotopic composition over the last couple of thousand years is a surprisingly difficult problem, and in many cases has circular dependencies on known ages of artifacts.
PTH doesn't really make an appearance because it's actually one of the less interesting theories. I don't believe these, but the facts the authors bring up to argue for their position are always interesting to learn about. Part 3 discusses some of the issues with 'scientific' dating methods and covers the circularity problem you discuss.
His criticisms of radiocarbon dating are interesting (while not completely supportive of his hypothesis). But it is an interesting rabbit hole to dig into; the relationship between radiocarbon dating and dendrochronology, and the attempt to get an accurate historical record of carbon isotopic composition over the last couple of thousand years is a surprisingly difficult problem, and in many cases has circular dependencies on known ages of artifacts.