The Crusades were not the primary vehicle for the transmission of Islamic learning to Europe. Just as today we wouldn't look to "grunt" soldiers as paragons of cross-cultural exchange, the Crusaders were not exactly in the Holy Land to hear about all the great things that had been happening in the libraries of Islamdom.

Rather, the primary route of such transmission was through Muslim Spain. Paper, for instance, reaches Europe via Spain and southern France, not through Byzantium. In the case of philosophy this knowledge or revival of interest in Aristotle came via Muslim philosophers whose commentaries and works of Aristotelian scholarship were translated from Arabic, often via Hebrew or I think in some cases Ladino, into Latin and were then distributed around western Europe.

The result was that by the time of Thomas Aquinas in the 13th century writing in his Summa Theologica he was referring to Averroes (Ibn Rushd) as the Commentator on the works of Aristotle and also included frequent citations to Avicenna (Ibn Sina).

I'm not familiar enough with the manuscript tradition of the preservation of classical philosophy to comment on that but my understanding is that, yes, medieval philosophers would have eventually been able to gain access to Greek originals, but the impetus to do so was based on the Islamic scholarship based above.

It's also worth noting that to the extent that these works were persevered in Byzantium in Greek it was largely based on the huge demand for these materials in Islamdom itself. So in the so-called Byzantine dark ages of the 7th and 8th century the study and copying of classical philosophic materials basically stops, and then seems to be re-ignited in close connection with the translation movement in Islam, though as far as I'm aware the relationship between these developments has not been fully established (e.g. did the Byzantines start copying this material to sell to an Islamic audience? Or did the Islamic demand just remind them 'hey, this stuff is worth keeping?')

Source wise:

For more on the translation movement see Greek Thought, Arabic Culture: The Graeco-Arabic Translation Movement in Baghdad and Early 'Abbasaid Society by Dimitri Gutas.

For cultural exchange during the Crusades, see my response HERE and /u/sunagainstgold 's response HERE, among other other similar posts.

Hopefully someone else can comment further on the manuscript tradition for Aristotle as it's not a topic I'm especially familiar with.

edit: typos