This post is one big triviality. But I only realised now why homotopy colimits in triangulated categories are defined the way they are defined.

Just think of triangulated categories as a bit wonky abelian categories, and in (non-wonky) abelian categories a colimit can be constructed from having coproducts and coequalisers. In a triangulated category you don't have coequalisers / cokernels, but cones play this role. Hence one takes a particular cone of a direct sum that simulates a coequaliser, and one is done.

Of course, everything dualises to homotopy limits.

Why did no-one tell me this?!

Each statement is to be interpreted under suitable conditions on the objects in sight, e.g. existence of direct sums etc. etc.