Was the Latin rule (Frankokratia) of the former Byzantine Empire the reason Greek texts and philosophy were so widespread (going to scholars such as Thomas Aquinas) or was the dissemination of Greek texts because of interaction with the Arabic world?
William of Moerbeke seems to have been the one who allowed for the translation of texts during the Latin rule of the Byzantine Empire. Wikipedia writes of him:
William of Moerbeke, O.P. (Dutch: Willem van Moerbeke; Latin: Gulielmus de Morbeka; 1215–35 – c. 1286), was a prolific medieval translator of philosophical, medical, and scientific texts from Greek language into Latin, enabled by the period of Latin rule of the Byzantine Empire. His translations were influential in his day, when few competing translations were available, and are still respected by modern scholars.
What do historians think?
I should note that you're overlooking the efforts of Christian monks in Europe itself, as pointed out by u/Veritas_Certum in their answer on a similar topic yesterday. u/XenophonTheAthenian also addresses just how some texts came to Western Europe, and I especially recommend the discussion beneath that post. William of Moerbeke certainly played his part in bringing Greek texts out west (both linked threads mention him, in fact) - but he's far from the only one.
And of course, if anyone else would like to address this topic in their own words, please don't let this post stop you! More posts are always, always welcome.