Well in a way is about politics but more in a Geo political sense. There were Kings and commanders on the European side that either wanted to gain favor with the pope and the Roman Catholic Church so as to be granted more power and possibly install one of their own as Pope.

Or they wanted to be the one to claim victory and use that as means to separate themselves from the pope and the RCC and from there just as Martin Luther and eventually Henry the 8th be able to create their own branch of the church and be able to run their country’s as they wished.

On the moorish/Muslim side of the crusades it was about power and the rights to the land for political clout amongst Islamic world. To be the one hold all of the key sites of the Islamic faith as well as remove all of Judaism so as to declare Ishmael as Abraham’s one true son would give that leader immortality and his sons everlasting control over the Islamic faith.

So you’re right in the sense that it was never about the inhabitants of Jerusalem but neither the inhabitants of Europe they were merely pawns in a game of kings, sultans and popes.