It's very hard to separate Annales School from the personalities and times, and I would argue impossible. For medievalists, we contend with a large body of influential work produced by this 'school': Bloch, Braudel, Le Goff, Duby, Ladurie. But even so, it would be hard for me to quantify the 'effect' of this school in any other way than ideological. From Bloch's Feudal Society to Ladurie's Montaillou we are confronted with history as told from the point of view of peasants, or so-called lower orders of medieval society. This is a radical departure from 19th century names-dates-of-important-people historiography, and it presented a face to medieval history that matched 1960's counter-culture which was the Annales' coming of age.

The stories they tell are fantastic and well told, and latter-day Annaliists like Duby and Ladurie were strong personalities which suited them to television and mass media - marking out part of the basis of their influence in the 70s and 80s and the marketability of their books in America.

Certainly late 20th century Medieval historiography is profoundly changed by Duby's 1950's study of feudal relationships in the Maconnais (in its exceptionally attenuated deep research of primary sources) and Ladurie's Montaillou (as a model of micro history). These extend from Bloch's historiographic philosophy expressed in his The Historian's Craft: the historian's task is to look at structure not narrative, and the historian must realize they are looking at 'traces' of peoples, not facing those historical people directly. This opens the gates to bringing in various non-traditional sociological, cultural, psychological analysis to create a greater 'understanding' of the subject culture and not just a few people. It permits great latitude for the historian's imagination, particular where evidence is thin, and to a great degree this has been very influential on modern historians where we try to fill in gaps of records.

From the point of view of methodology, various Annales texts have their faults: they are selective in their outlook; they overlay a lot of willy-nilly 'Durkheimian' sociological analysis almost to the point of anthropology but without the science; they are often uncritical of their sources - or uninterested in interrogating the biases of the source they work with, a must certainly in medieval studies where a single differing interpretation of a latin word in a document can send historiography in a different direction.

Other than the broadest of generalizations, I would have a hard time extending a critique of this so-called school on the basis of a unifying idea. Braudel's faults are not the same as Le Goff's, and they aren't the same as Bloch's.

I'm not sure there is an Annales methodology anymore - it's been subsumed into, or perhaps over-taken by, the cross-fertilization of disciplines in the academy and within general culture. It doesn't seem so radical or different anymore.

As for the claims to 'total history', that's Braudel marketing and it's very a French mid-20th century expression.

Edit: some further comments about Bloch