This entire question is troublesome to me. Where to begin?

Relligion is incredibly useful for rulers to unify culturally similar peoples under a central "holy" authority.

An incredibly cynical view of religion that mistakes the means for the ends. At any rate, it's impossible to know if this was Muhammad's motivation for spreading the new religion, or he was a genuine believer in his words.

It seems particularly baked into Islam,

It does?

Having been besieged and conquered as a prized city by Babylonians, Romans and Persians is there any causality to this timeline:

You are talking about events that were hundreds of years apart for the most part, aside from the Jewish Revolts under Rome. For most of it's history under Rome, Jerusalem was at best a relatively small backwater after it's sacking by Rome, and it remained this way for most of the Byzantine period as well.

There was no grand design or plan to conquer Jerusalem as a built in goal for Islam.

I'll echo Spoonfeedme and say the entire question is quite troublesome. It's loaded with premises that are patently false.

There's no debate that religion is an incredible unifier and will bring together disparate people who otherwise have nothing in common. However, your question seems to indicate that you believe Muhammad saw himself as a unifier of Arabic culture. This is not true. There are verses of the Qur'an which describe Muhammad as being sent to mankind. There are hadith which describe him as being the "leader of the children of Adam." The early Muslims quite evidently envisioned Islam as a global phenomenon and a unifier of the world, not just Arabs.

Your second paragraph is quite misleading. The "conquest" of Makkah was hardly an attack. By that time, Makkah's elite had long since withered away. They had either died in battle and left no one of a similar caliber to replace them (Abu Jahl, Waleed ibn Mughira, Utbah ibn Rabi'ah) or had themselves become Muslim (Khalid ibn Walid, Amr ibn Al As). When Muhammad showed up with 10,000 people, the Makkans literally put down their weapons and said "we surrender." Literally. Abu Sufyan met with Muhammad the night before and said he would put up no resistance and to march right in. There simply was no way Makkah could muster anywhere near the manpower to fight a fraction of the size of the Muslim army.

Anyway, to address the question about Jerusalem, it's very clear that the early Muslims saw themselves as having a claim to it. The very first direction of prayer was towards Jerusalem. It is mentioned in the Qur'an as having the "farthest mosque" and that the lands around it have been made sacred. Muhammad is recorded to have said "God bless us in our Madinah and our Jerusalem." He is also recorded to have said that no religious pilgrimage can be made to any location other than Makkah, Madinah, and Jerusalem. There is no doubt that the early Muslims saw themselves as eventually having to march towards Jerusalem and conquer it (which they did during the tail end of Abu Bakr's caliphate).