There is no doubt that Greek scholars had a profound influence on the Renaissance in Western Europe. But you're vastly over-stating the influence of the fall of Constinople on that impact, just as you're vastly overstating the military and economic potential of the "Empire" in 1444.
Greeks had been instrumental in re-examining and re-introducing classical works for centuries, before 1453. But this rarely eminated from the Capital, which was still often hostile to the pagan aspects of Hellenism; for example, they tended to adhere to a strict preference for Aristotle over Plato, as did the Latin church. "Humanists" were often viewed with suspicion in Constantinople, as in Rome. Indeed, it was partially because of the weaknesses of the "Empire" that scholars were more inclined to study less appropriate classics in places like Ottoman controlled Adrianople or semi-independent Mistra. But the conduits for Greek influence had been active for centuries from places like Sicily, Southern Italy, Venice/Genoa with their holdings, Latin Crusader states, inter-marriage among prominent Greek and Italian families, etc. And this could be a two-way street - ties between Florence and Mistra, for example, undoubtedly influenced both Greek and Italian speaking scholars in these centers.
Whether there was any particular boost in this ongoing process as a result of 1453 "refugees" is highly debatable. Despite widespread repetition, there isn't much evidence of any sudden influx in 1453. On the contrary, there is a counter argument that in the immediate aftermath, the flow of intellectual talent from Greek speaking areas to the west slowed as the Sultan demonstrated an unusual eagerness to rebuild the city, including its intellectual life, and tried various means to lure and accommodate scholars, with some apparent success. Not all Sultans were so accommodating, however, and the 16th century has ample evidence of a return to the pattern of some scholars preferring to study/work in Venetian territory or the west, rather than doing so in Ottoman territory.