Unfortunately, science journalism tends to fail to capture the role that books like this play in academic history. I doubt that the author says that climate was a leading factor in Rome's fall, but I'm sure he makes a strong argument that we should not neglect its role.

It's important in the sense that it investigates a typically neglected dimension, but I think historians will be hesitant to rank factors that led to the fall. Sufficiently self-aware historians are aware that the etiology of past events is a tricky business, and the best one can do is discuss a sequence of events with great detail that may or may not be linked by causation.