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.
St. Mark's is a good-sized church, but if it truly represented the despoiling of the entire city of Constantinople then Constantinople had next to nothing in it.
I've also been to the Vatican museum and there is a decent-sized collection of Byzantine art and "treasure" - again nowhere near enough to represent the despoiling of an entire city.
Additionally, the Vatican received absolutely nothing at all from the sack of Constantinople because the Pope excommunicated both the Western generals responsible for the sack.
The Western sackers were privateers acting on their own initiatives for their own private enrichment - the Vatican specifically instructed them not to make war on the Byzantines.