The point of departure is the following : There is a very simple way to construct stable homotopy categories for small categories by forming a category of fractions (for example, used by Higson in his construction of E-theory). On the other hand, to construct such categories in a more general setting, one uses triangulated categories or the Spanier-Whitehead construction.

Now in principle, if one just uses categories of fractions, one does not get that the homomorphisms between fixed objects form a class. Why exactly is this a problem?