I think it is not good point of view to think that $A \mapsto H^0(Mod-A)[Qis^{-1}]$ is canonical construction for dg-algebras and $\mathcal{C} \mapsto H^0(Mod-\mathcal{C})/H^0(Acyc-\mathcal{C})$ is canonical construction for dg-categories. In fact, both constructions can be done for algebras and for categories in the same way. Third very useful model for $D(\mathcal{A})$ is $H^0(SF-\mathcal{A})$ - homotopy category of all semi-free dg-modules-$\mathcal{A}$ which just strictly full triangulated subcategory of $H^0(Mod-\mathcal{A})$. So I see there are 2 questions:

1. Let $A$ is dg-algebra, why $D(A)$ and $D(Mod-A)$ canonically triangulated equivalent?
2. Let $\mathcal{C}$ is dg-category, why $H^0(Mod-\mathcal{C})[Qis^{-1}]$ and $H^0(Mod-\mathcal{C})/H^0(Acyc-\mathcal{C})$ canonically triangulated equivalent?

Both questions are quite classical. 

Let $F : A \to Mod-A$ canonical embedding of $A$ (which we consider as category with one object $*$ and $End(*) = A$ to $Mod-A$ as rank 1 free module-$A$).

First question is composition of following facts

1. Let $SF-\mathcal{A} \subset Mod-\mathcal{A}$ full dg-category which consists all semi-free dg-modules. Then induced localization functor $H^0(SF-\mathcal{A}) \to D(A)$ is quasi-equivalence. (Every dg-module-$\mathcal{A}$ have semi-free resolution). [Keller Deriving DG-categories 3.1][1]

2. Let $G : \mathcal{B} \to \mathcal{C}$ be full embedding of dg-categories. And let $G^* : SF-\mathcal{B} \to SF-\mathcal{C}$ be the extension dg-functor. Then the induced functor $H^0(G^*) : D(\mathcal{B}) \to D(\mathcal{C})$ is fully faithful. If, in addition, the category $H^0(\mathcal{C})$ is classically generated by $Ob \mathcal{B}$ then $H^0(G^*)$ is an equivalence. Proposition 1.15 in [Luntz Orlov, "UNIQUENESS OF ENHANCEMENT FOR TRIANGULATED CATEGORIES"][2]

3. $F$ is full embedding of classical generator of $SF-\mathcal{A}$. It is quite obvious, you can construct any semi-free modules from free using cone operation. Argument of such type must be also written in [Keller Deriving DG-categories][1]

Your second question is more or less also implies from theorem 3 in [Keller Deriving DG-categories][1].


  [1]: http://www.mi-ras.ru/~akuznet/dgcat/Keller%20Deriving%20DG%20categories.pdf
  [2]: https://arxiv.org/pdf/0908.4187.pdf