I'll respond to each of these points in the same order as they were mentioned.

1. Illyrians did not inhabit "most of mainland Greece" during Alexander's lifetime. This picture shows a map of Greece and its surroundings in 336 BC, the year Alexander became king: https://en.wikipedia.org/wiki/Glaucias_of_Taulantii#/media/File:Map_Macedonia_336_BC-en.svg. Illyria is not shown specifically, but it's essentially the land along the coast of the Adriatic Sea, today mostly part of Albania and part of Croatia. In this period, northern Greece was inhabited mostly by Macedonians (who despite being seen as semi-barbarous by the Greeks of the Peloponnese still spoke the same language and worshiped the same gods), while the Peloponnese was inhabited by, well... Greeks. Obviously in the 4th century BC national censuses didn't exist so we can't say how many Illyrians 'migrated' east and south into Greece, but it would be a very low number if it happened at all; language and cultural differences would be too great, and this type of movement of peoples did not usually happen, since the vast majority of the population were peasants who needed to work on their farm to survive. Just packing up and moving to a place already inhabited by other people wasn't a great idea.
2. Alexander was not Illyrian. He was born in Pella, the capital of the ancient Kingdom of Macedon, and which is today part of northern Greece. His father was Macedonian and his mother Epirot - while much of Epirus is now part of Albania, in this era Epirus was linguistically, culturally and religiously almost identical to the Greeks of the Peloponnese. Similarly, as mentioned above, Macedonians in this era were also far closer to the Greeks of the Peloponnese than the usage of that name suggests in the modern-day. While Alexander personally disliked the Greeks and would identify solely as Macedonian, he was still far closer to being Greek than being Illyrian in every regard, despite the tendency of Greeks of this era to lump everyone north of the Peloponnese into the general category of "barbarians".
3. There is no evidence to suggest that "Illyrians made up more than half of Alexander's army". Fortunately we have a very good source on Alexander's military campaigns, in The Campaigns of Alexander by Arrian, a 2nd century Roman writer who was born in Greece. He describes Alexander's army - the one that he invaded the Persian Empire with - as being an army roughly 40,000 men-strong, and being almost entirely Macedonian (with important exceptions such as having contingents of Cretan archers and Rhodian slingers). Alexander took as many Macedonian soldiers with him as he could while also leaving enough at home to defend the country from the barbarian tribes of the north (the Illyrians and Thracians) and the Greeks of the south. Despite nominally being vassals of the Macedonians, Alexander trusted neither the northern tribes nor the Greeks, and had to suppress rebellions in 336-5 of the Thracians, Illyrians and Thebians. I find the claim your father made especially hard to believe since immediately upon becoming king the Illyrian tribes rebelled and Alexander had to subdue them with force. This would hardly have happened if "more than half" of his army was Illyrian. Alexander's army remained almost purely Macedonian until the very end of his life, when he started to recruit Persians into the ranks to make up for losses of Macedonian soldiers - an extremely controversial move. Alexander even sent back troops that the Athenians and other Greek city-states had sent as reinforcements - he didn't trust any soldiers except Macedonians.

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Sources:

The Campaigns of Alexander by Arrian

Alexander of Macedon by Peter Greene