For example, I am referring to:
Several high-ranking Persian nobles were killed by Alexander himself or his bodyguards, although Alexander was stunned by an axe-blow from a Persian nobleman named Spithridates. Before the noble could deal a death-blow, however, he was himself killed by Cleitus the Black at Grannicus.
Alexander then mounted his beloved horse Bucephalus, took his place at the head of his Companion cavalry, and led a direct assault against Darius. The horses that were pulling Darius' chariot were injured, and began tossing at the yoke. Darius, about to fall off his chariot, instead jumped off. He threw his royal diadem away, mounted a horse, and fled the scene at Issus.
Alexander, while leading the charge, formed his units into a giant wedge, which quickly smashed right into the weakened Persian center. Darius' charioteer was killed by a spear, and chaos rang out as everyone (incorrectly) thought it was Darius who had been killed. The Persian line then collapsed, and Darius fled. Darius escaped with a small core of his forces remaining intact, although the Bactrian cavalry and Bessus soon caught up with him. The remaining Persian resistance was quickly put down. In all, the Battle of Gaugamela was a disastrous defeat for the Persians, and possibly one of Alexander's finest victories.
Pretty much in every single battle Alexander and his cavalry make some heroic charge and directly go to the heart of the Persian army and directly towards Darius. It always makes out Darius to be weak willed and a coward as well while Alexander is elevated to Homeric heroism.
How true is all of this, and is their a consensus among historians? From what I understand Hans Delbruck pretty much states that all of this is Macedonian propaganda.
There is a lot of truth in this. Often kings in that era led their men into battle, and Alexander and his father Phillip rode with their companion cavalry which agrees with your source.
Lets look at the war battles were fought. The lines would come together after skirmishing for a while and fight until one side broke and ran and it was then the victorious cavalry's job to chase down the remnants of the army. Alexander and his companion cavalry certainly would make a try for the king like they did at Issus. This makes sense because if Alexander captured Darius, it would have meant he would win the war almost immediately.
So in conclusion I think that while maybe embellished a little, the accounts seem historically plausible because they align with the customs of that time. I don't see any reason to believe that Alexander didn't do any of those things.
Sources: The Life of Alexander