Why is the triangulated category of motives easier than the abelian one? There are several expository articles with the title "You could have invented [insert something mysterious here]" (a notable one being about spectral sequences, possibly it even started this genre). This question is somewhat similar in spirit to them. 
Here it is stated that "Deligne first suggested that it might be easier to define the derived category $DM(S,\mathbb{Q})$ of the hypothetical abelian category of mixed motives." First time I heard about this, it seemed a little bit counterintuitive to me. We were doing some abelian stuff, why should passing to the derived category make anything any simpler? Of course, it is easy now to point to the success of Voevodsky and others and say that it was totally obvious. 
The question is assuming you never heard about Voevodsky, Morel, etc., you are in the 1960's, how could you arrive at the idea that the triangulated category is easier to construct than the abelian category of mixed motives?
 A: Existence of the abelian category of mixed motives over a field makes (extremely hard and widely open) Beilinson-Soulé vanishing conjecture to be manifestly true: it follows from the fact that $$Ext_{MM}^i(\mathbb{Q}(0),\mathbb{Q}(n))$$ vanishes for $i<0$.
 On the other hand,  this does not need to be true in the  triangulated category of mixed motives.
A: I believe that there is more than one answer to this question.


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*Even if you don't know anything about the success of Voevodsky and others, you may start with a certain "category of complexes of varieties" and then "impose relations" by localizing by certain elements. Then you will obtain a certain triangulated categories such that certain "relations" (well, basically homotopy invariance and certain Mayer-Vietoris ones; it is more difficult to get etale descent and especially Gysin triangles using this method) that has certain "realizations". On the other hand, to obtain an abelian category motives you should "enforce" certain long exact sequences (of symbols of the form $H^i(V)$), and I doubt that any general formalism allows this.

*Certainly, Deligne was and is extremely smart, but I doubt that his original idea was really "prophetic". As far as I remember, he actually proposed to look at "compatible systems of realizations" similar to that considered in books of Jannsen and Huber. So, this a certain extension of the idea of mixed Hodge structures, and one can probably explain "explicitly" why mixed Hodge complexes are "better" than complexes of mixed Hodge structures. 
