Constantinople was sacked by an army consisting of various mercenaries, Venetians and Byzantines. The largest contingent among the troops that sacked Constantinople were themselves Byzantines sacking their own capital city.
The sack of Constantinople occurred in 1203, 250 years before Constantinople fell to the Turks. Pinning the military weakness of 1453 on an incident from 1203 is preposterous.
Most of the treasures of Constantinople were gone well before the Venetians and their allies took the city. Isaac II had looted the treasury before he left the city 8 years earlier, and his successor melted down both Catholic statuary and Orthodox icons for the gold he needed for bribes to keep his tenuous hold on power.
Very little "treasure" was hauled away from Constantinople since very little was left in a city that had been torn by civil strife and kleptocracy for at least a decade.
A good deal of the "Treasure" looted from Constantinople was the sons and daughters of the city.
The sack of Constantinople by a supposed Crusader force was a disgrace. It helped weaken the city and set it up for later Muslims conquest.
According to a recent book on that subject, a series of incredible blunders, fed by greed, caused that action. And the Pope actually excommunicated the leaders of the Crusade when he heard what they had done. Capturing Constantinople was not the objective of the Latin Church nor of the rank and file Crusaders but was the creation of the Venetians and the Crusade's inept leaders.
I don't think you are right. The old treasury Building and the Church of St. Marks in Venice is loaded with stuff from Constantinople, as is the Vatican.