The main reason was his liberal policies towards the native population. A few examples are that he allowed the Carians to be ruled by their Queen, gave certain tax freedoms to city-states that accepted him peacefully in Asia Minor, allowed a certain level of autonomy for the Egyptians and let Besus remain satrap of Babylonia. He was careful when choosing the generals who would act as satraps and the gentle approach of most of them (for example Peucestas in Persia) made them quite famous and loved by the indigenous people. This also allowed the Macedonians to recruit thousands of foreign troops who even after Alexander's death remained completely loyal to the crown. Without those troops, the Macedonians would have been unable to hold the Empire together. Alexander and his generals tried not to offend the locals, payed respect to their gods and allowed them to retain a certain extend of autonomy. They tried to backed up their morality behind the invasion by proclaiming that they wished to free the people from the Persian tyrannical rule and rightfully avenge the invasion of Greece by the Persians. Added to that, there were also very finely measured and planned outbursts of violence, for example against Persepolis, in order to make it clear to the natives that Alexander is no fool and that he will destroy those who stand in his way. The destruction of Persepolis also served as the peak of the 'vengeance policy' meaning that it was the answer to the burning of Athens.
As for what Constantius said, I completely disagree, Alexander's personality was essential for the creation of the Empire but no so much for its survival. Most of his ideas and policies had already been adopted by his generals and successors like Ptolemy, Seleucus and Eumenes of Cardia. People usually argue that the Alexandrian Empire was torn to pieces immediately after Alexander's death and they ignore certain facts. In just 11 years, Alexander had carved a tremendous dominion which encompassed hundreds of tribes, dozens of different religions and so on. Yet, the dynastic crisis following Alexander's death was avoided by the Partition of Babylon which reorganized the Empire and allowed it not only to keep itself together, by avoiding rebellions of subjects whose numbers far exceeded those of the small Macedonian army but also to expand in Cyrenaica and Cappadocia. In the process, the Macedonians also managed to crush the Greeks in the Lamian War, the Greek mercenaries who revolted in Bactria and an invasion of Scythians in Thrace. The division of the Empire can be attributed to certain political figures, such as Antipater and mostly, Antiochus who sparked distrust and hate between the Diadochi. Up until a certain point, the Alexandrian Empire was holding up well and it seemed as Alexander IV would eventually become the rightful King.