I recommend "Augustus: First Emperor of Rome" by Adrian Goldsworthy.
Given that "democratic" didn't mean the same thing then it does now (with suffrage limited to a small group of the uber-rich), and that some of the problems he was fixing was "the threat of this army I happen to have" and "this war I actively participated in", I don't think it is wrong. He wasn't in Rome when the Senate awarded him power and the Vestal Virgins drank in his name, which isn't something that would be commanded.
After decades of war and strife and food shortages, peace under one warlord looked more appealing than having three who would likely eventually be at each other's throats.
Well, first, you have to consider that the Roman republic was never really democratic but was at the hand of a small aristocracy which had sometimes but rarely the interest of the Romans at all, let alone the non citizen inhabitants.
But even, then it is definitely not accurate. Augustus gained powers through the numerous conflicts which followed Caesar murder at a time when the Republic was already challenged thanks to legions he more or less inherited (oversimplification) from Caesar. He was given powers by the Senate through what we would call rubber stamping only after his military power was inescapable.
