A&R make their arguments by historical narrative, although underlying these, there are often academic papers that use the toolkit of modern econometrics. This is a great approach – a credible historical narrative is much more convincing than a bunch of cross-country regressions. And, looking far back into history is surely the correct thing to do. For to answer the question above, you must think both about contemporary variation (why are some nations poor now and others rich now) and about historical variation – why were all nations poor for most of recorded history, before the take-off into growth of Britain, then Europe, then the world in the past 3 centuries.
These guys are not professional historians by training. This creates a bit of a trust problem. When I read their (very interesting) stories about Ethiopia, the French Revolution, Indonesia, etc. I wonder to myself how many specialist historians of the time and place would agree. Their narratives are plausible, but I can often imagine alternative stories.
One episode where I do know a little is the decline of the Roman empire. Here A&R’s story seems pretty unusual. They blame the end of the Roman republic for the decline of the empire, which collapsed four centuries later in the West and fourteen centuries later in the East. In fact much of the empire’s expansion happened under the emperors. They have no persuasive causal link between political institutions and the overtaxation which may have caused the decline of the Roman countryside in the third century. (Here’s a modern book which gives lots of detail, without losing track of the big picture.)