I don't really think that triangulated categories are abominable, but they certainly have their problems which are a result of having forgotten the higher homotopies. For instance, non-functoriality of mapping cones can be fixed via dg-enhancement.
Another problem has to do with localization for triangulated categories. The fact that one can put a model structure on a suitable category of dg-categories and take homotopy limits for instance is a very useful thing and allows one to talk more meaningfully about gluing and working locally. One can do this in the derived category but only in quite simple situations and it requires extra structure for it to work well (e.g. the structure of a rigid tensor category compatible with the triangulation).
At least one good place to read in more detail about these problems and how to fix them using dg-categories is To¨en's Lectures on DG-categories which can be found here.
I haven't really said anything about the A-infinity point of view, but I don't know it so well yet - hopefully someone else can say something. But I do know that at least one of the same problems rears its head namely that of wanting to work locally. For instance in homological mirror symmetry one would like to glue Fukaya categories of complicated things together from those of easier things. To do this one would certainly need to use the A-infinity point of view before taking derived categories and if it works it should be because one can take the appropriate homotopy limit (does the model structure to do this actually exist by the way?).
