Let $\mathcal A$ be an abelian category. Your condition (zero map on cohomology implies null-homotopy) is equivalent to the condition that $\mathcal A$ be semisimple (i.e. any exact triple $0\to X\to Z\to Y\to 0$ in $\mathcal A$ splits, i.e. is isomorphic to $0\to X\to X\oplus Y\to Y\to 0$).
On the one hand, your condition implies all Yoneda Exts are zero, so that the homological dimension of $\mathcal A$ is zero; i.e. $\mathcal A$ is semisimple.
On the other hand, if $\mathcal A$ is semisimple, the functor $h:(K^n,d^n)\mapsto (H^n(K),0)$ which sends a complex $K$ to its complex of cohomology objects with zero differentials in all degrees, induces an equivalence between $D(\mathcal A)$ and the category of cyclic complexes of objects of $\mathcal A$ (complexes with zero differentials in all degrees).