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.