I would propose a different alternative. It is the theory of (Grothendieck's) derivators, at least the stable variant. It was also developed by Heller under the name "homotopy theories" and very much related to Keller's "towers of triangulated categories" and to Franke's "systems of triangulated diagram categories". Roughly, to a base triangulated category one adds all homotopy limits and colimits, essentially adjoints (that arise as Kan extensions) to the constant diagram with values in a triangulated category of "coherent diagrams".
Once you have a derivator, to be stable is a property, not a structure on it. This property is reasonably easy to check in the main examples and distinguishes stable phenomena. Stability immediately yields a collection of distinguished triangles satisfying the usual axioms. Also octahedra and higher triangles are produced by this property and they behave in a right way form the homotopical point of view, implicitly satisfying the universal properties up to homotopy that defines them.
A very nice exposition is the paper by Groth "Derivators, pointed derivators, and stable derivators" (Algebraic & Geometric Topology 13 (2013), 313-374)
For some people this notion is simpler than $\infty$-categories and encompasses the work recently done with just the axioms mentioned by Peter May in his answer together with the existence of arbitrary coproducts.
The idea of Grothendieck was to express the deep meaning behind the notion of homotopy. To the extent he achieved this is debatable. But, the flexibility of stable derivators for extending homotopical constructions in triangulated categories without making recourse to model categories is one of the features some people may find useful.