I would subtitle this question "Bourbaki's dream." The dream faltered on the foundations. Bourbaki tried to give a half-baked half-formalized naive set theory resulting in an embarrassment of epic proportions (with regard to their volume Theory of Sets; of course other volumes have been extremely successful, like the Lie theory volume) that has been detailed by Adrian Mathias, an expert in the field unlike any of the Bourbaki, in a series of recent detailed critiques (not merely his essay The ignorance of Bourbaki). What I am trying to suggest is that checking all of the previous results will get you hopelessly bogged down in the foundations.
In the comments below, some of the editors requested evidence that work by Grothendieck was blocked by the Bourbaki. Here is a first sample from a 1992 paper by Corry:
Eilenberg himself was commissioned several times with the preparation of drafts on homologies and on categories, while a fascicule de résultats on categories and functors was assigned successively to Grothendieck and Cartier. However, the promised chapter on categories never appeared as part of the treatise. As we shall see in greater detail in the next section, the publication of such a chapter could have proved somewhat problematic when coupled with Bourbaki's insistence on the centrality of structures. The task of merging both concepts, i.e., categories and structures, in a sensible way, would have been arduous and unilluminating, and the adoption of categorical ideas would have probably necessitated the rewriting of several chapters of the treatise. This claim is further corroborated by the interesting fact that when the chapter on homological algebra was finally issued (1980), the categorical approach was not adopted therein.
My perception is that the Bourbaki concept of structure, while in principle similar in spirit to category theory, was hopelessly tied in with their naive set-theoretic realism, and as a consequence created an impediment of the sort detailed by Corry.
While I am in favor of systematization, the Bourbaki project to get to an alleged bottom of all of mathematics is more of a collectivization than a systematization, and is disturbingly procrustean.