Here is a short argument why we don't expect generalized cohomology theories to behave so well.
In the stable homotopy category, there is a generalized homology/cohomology theory represented by the sphere spectrum S, so that S_*(X) are the stable homotopy groups of X. It has a multiplication-by-2 self-map and we can use the triangulated structure to find an exact triangle S -> S -> M -> S. We think of M as "the sphere mod 2", and it is called the mod-2 Moore spectrum.
In most derived categories coming from algebra, such an exact triangle would have the property that the multiplication-by-2 map on M was zero. However, we know that this is not the case here; the multiplication-by-2 map is not zero, but the multiplication-by-4 map is.
One of the problems with using a "functor" language to get at generalized cohomology theories is that given a natural transformation E -> F of generalized cohomology theories, it is not clear what the associated cofiber should be in order to produce a triangulated structure. Spectra have a natural triangulated structure and they rectify this problem. (A map between spectra also includes "phantom" data that isn't easily detected by the natural transformation between associated generalized cohomology theories.)