Here is a previous answer from /u/xenophontheathenian, which discusses this question in the reverse (ala 'Why didn't Alexander go north,) and explains a bit about this topic. Most relevant to your question is the discussion of Macedon's hegemony over its Greek subjects/allies, and Philip's development of the army that you touched upon. The 'Panhellenic' movement for a war against Xerxes was very popular among the Greek nobles, and formed a key part of Philip's claim to legitimacy as leader of the Greek coalition, aka the League of Corinth. /u/xenophontheathenian identifies Isocrates as a key player in the intellectual movement behind war with Persia, that would go on to prop up Phillip's regime. Especially near the end of Phillip's life, war with Persia was the public purpose of both Phillip's domination of Greek states, as well as the development of his army. Alexander inherited both his father's Kingdom and his army, but their continued coherence was in many ways premised upon future war with Persia. Alexander also received tutelage from many Greek notables as part of his induction into Hellenic high society, and could have developed a genuine belief in the Panhellenic movement due to growing up with these entourages, although that is speculation on my part.

There is much more to be said on this subject, and there might even be more recent/pertinent answers from from uXenophon or others. I encourage others to contribute new answers and insights.