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A novel approach for analyzing "classical" alternatives to quantum mechanics for explaining the statistical results of an EPRB-like experiment is proposed. This perspective is top-down instead of bottom-up. Rather than beginning with an inequality derivation, a hierarchy of model types is constructed, each distinguished by appropriately parameterized conditional probabilities. This hierarchy ranks the "classical" model types in terms of their ability to reproduce QM statistics or not. The analysis goes beyond the usual consideration of model types that "fall short" (i.e., satisfy all of the CHSH inequalities) to ones that are "excessive" (i.e., not only violate CHSH but even exceed a Tsirelson bound). This approach clearly shows that noncontextuality is the most general property of an operational model that blocks replication of at least some QM statistical predictions. Factorizability is naturally revealed to be a special case of noncontextuality. The same is true for the combination of remote context independence and outcome determinism (RCI+OD). It is noncontextuality that determines the dividing line between "classical" model instances that satisfy the CHSH inequalities and those that don't. Outcome deterministic operational models are revealed to be the "building blocks" of all the rest, including quantum mechanical, noncontextual, and contextual ones. The set of noncontextual model instances is exactly the convex hull of all 16 RCI+OD model instances, and furthermore, the set of all model instances, including all QM ones, is equal to the convex hull of the 256 OD model instances. It is shown that, under a mild assumption, the construction of convex hulls of finite ensembles of OD model instances is (mathematically) equivalent to the traditional hidden variables approach. Plots and figures provide visual affirmation of many of the results.