Suppose I start with an abelian category $\mathcal{A}$, form its category of complexes $C(\mathcal{A})$ and consider a short exact sequence in this category:
$$0 \to A^{\bullet} \to B^{\bullet} \to C^{\bullet} \to 0 $$
If this sequence is termwise split (that is for all $n$ the sequence $0 \to A^{n}\to B^{n} \to C^{n} \to 0$ is split) then the triangle associated to this sequence is a distinguished in $K(\mathcal{A})$. (See pg. 28 - 30 of this Stacks project article).
My question is, can we form a triangle from the above sequence if it is not split?
If yes, what relation, if any, does it have to the distinguished triangles in $K(\mathcal{A})$?
If no, why is it so commonly said that the distinguished triangles take the place of short exact sequences, when to me it would seem that they are only associated to a particularly simple kind of short exact sequence?