I learned the following partial answer from Peter Haine (any errors are of course my own). In the following I will ignore any set-theoretic issues which may arise.

Let $\mathsf{A}$ be an abelian category, with derived category $\mathsf{D}(\mathsf{A})$ and derived $\infty$-category $\mathcal{D}(\mathsf{A}),$ and recall that $\mathsf{D}(\mathsf{A})$ is the $1$-categorical localization of $\mathsf{Ch}(\mathsf{A})$ with respect to the class $\mathsf{qis}$ of quasi-isomorphisms, and that $\mathcal{D}(\mathsf{A})$ is the $\infty$-categorical localization of $\mathsf{Ch}(\mathsf{A})$ with respect to $\mathsf{qis}.$

If $[1]$ denotes the walking morphism, and $\mathsf{I}$ is the walking isomorphism, one model for this localization is the pushout of $$\coprod_{f\in\mathsf{qis}} \mathsf{I}\leftarrow \coprod_{f\in\mathsf{qis}} [1] \to \mathsf{Ch}(\mathsf{A}),$$ formed either in the $\infty$-category $\mathcal{C}\mathsf{at}_1$ of $1$-categories (for $\mathsf{D}(\mathsf{A})$) or the $\infty$-category $\mathcal{C}\mathsf{at}_\infty$ of $\infty$-categories (for $\mathcal{D}(\mathsf{A})$), where in the latter we interpret $1$-categories as $\infty$-categories via their nerves.

If we write $\mathrm{N} : \mathcal{C}\mathsf{at}_1\to\mathcal{C}\mathsf{at}_{\infty}$ for the inclusion of $1$-categories into $\infty$-categories, then $\mathrm{N}$ has a left adjoint, the homotopy category functor $\mathrm{h}.$

We then obtain a map $$\mathcal{D}(\mathsf{A})\simeq\operatorname{colim}\left(\coprod_{f\in\mathsf{qis}} \mathrm{N}(\mathsf{I})\leftarrow \coprod_{f\in\mathsf{qis}} \mathrm{N}([1]) \to \mathrm{N}(\mathsf{Ch}(\mathsf{A}))\right)\to \mathrm{N}\left(\operatorname{colim}\left(\coprod_{f\in\mathsf{qis}} \mathsf{I}\leftarrow \coprod_{f\in\mathsf{qis}} [1] \to \mathsf{Ch}(\mathsf{A})\right)\right)\simeq\mathrm{N}\left(\mathsf{D}(\mathsf{A})\right)$$ by the universal property of the colimit, and the adjunction of $\mathrm{N}$ and $\mathrm{h}$ then provide the desired canonical morphism $$ \mathrm{h}\left(\mathcal{D}(\mathsf{A})\right)\to\mathsf{D}(\mathsf{A}). $$

We immediately see that this morphism is an equivalence, since $\mathrm{h}$ being a left adjoint implies it commutes with colimits, and $\mathrm{h}\mathrm{N}(\mathsf{C})\simeq\mathsf{C}$ (naturally!) for all $\mathsf{C}\in\mathcal{C}\mathsf{at}_1.$

Additionally, we obtain the desired result for $\mathcal{D}_{\mathrm{qc}}(X)$ (despite the fact that this is not the derived $\infty$-category of some abelian category in general) by noticing that the category $\mathcal{D}_{\mathrm{qc}}(X)\subseteq\mathcal{D}(X)$ is defined by a condition on cohomology (which $\mathrm{h}$ respects).

In fact this construction then provides a functor from ``relative $\infty$-categories'' (i.e., the $\infty$-category whose objects are pairs $(\mathcal{C},\mathcal{W}),$ where $\mathcal{C}$ is an $\infty$-category and $\mathcal{W}$ is a class of morphisms in $\mathcal{C},$ and whose morphisms $F : (\mathcal{C},\mathcal{W})\to(\mathcal{C}',\mathcal{W}')$ are functors of $\infty$-categories $F : \mathcal{C}\to\mathcal{C}'$ such that $F(\mathcal{W})\subseteq\mathcal{W}'$) to the $\infty$-category of $\infty$-categories.

The desired property that these functorial equivalences should be compatible with derived pullback of sheaves should follow from expressing the derived pullback as an appropriate functor of relative $\infty$-categories, and noting that there is functoriality of these pushouts, but I have to think about this aspect of the construction more.

If I've made any errors in my recollection of the argument I learned, or if there are technicalities or subtleties I'm not doing justice, please do let me know. I'll leave the question open for now since the question of compatibility with derived pullbacks hasn't been completely answered yet.

I learned the following partial answer from Peter Haine (any errors are of course my own). In the following I will ignore any set-theoretic issues which may arise.

Let $\mathsf{A}$ be an abelian category, with derived category $\mathsf{D}(\mathsf{A})$ and derived $\infty$-category $\mathcal{D}(\mathsf{A}),$ and recall that $\mathsf{D}(\mathsf{A})$ is the $1$-categorical localization of $\mathsf{Ch}(\mathsf{A})$ with respect to the class $\mathsf{qis}$ of quasi-isomorphisms, and that $\mathcal{D}(\mathsf{A})$ is the $\infty$-categorical localization of $\mathsf{Ch}(\mathsf{A})$ with respect to $\mathsf{qis}.$

If $[1]$ denotes the walking morphism, and $\mathsf{I}$ is the walking isomorphism, one model for this localization is the pushout of $$\coprod_{f\in\mathsf{qis}} \mathsf{I}\leftarrow \coprod_{f\in\mathsf{qis}} [1] \to \mathsf{Ch}(\mathsf{A}),$$ formed either in the $\infty$-category $\mathcal{C}\mathsf{at}_1$ of $1$-categories (for $\mathsf{D}(\mathsf{A})$) or the $\infty$-category $\mathcal{C}\mathsf{at}_\infty$ of $\infty$-categories (for $\mathcal{D}(\mathsf{A})$), where in the latter we interpret $1$-categories as $\infty$-categories via their nerves.

If we write $\mathrm{N} : \mathcal{C}\mathsf{at}_1\to\mathcal{C}\mathsf{at}_{\infty}$ for the inclusion of $1$-categories into $\infty$-categories, then $\mathrm{N}$ has a left adjoint, the homotopy category functor $\mathrm{h}.$

We then obtain a map $$\mathcal{D}(\mathsf{A})\simeq\operatorname{colim}\left(\coprod_{f\in\mathsf{qis}} \mathrm{N}(\mathsf{I})\leftarrow \coprod_{f\in\mathsf{qis}} \mathrm{N}([1]) \to \mathrm{N}(\mathsf{Ch}(\mathsf{A}))\right)\to \mathrm{N}\left(\operatorname{colim}\left(\coprod_{f\in\mathsf{qis}} \mathsf{I}\leftarrow \coprod_{f\in\mathsf{qis}} [1] \to \mathsf{Ch}(\mathsf{A})\right)\right)\simeq\mathrm{N}\left(\mathsf{D}(\mathsf{A})\right)$$ by the universal property of the colimit, and the adjunction of $\mathrm{N}$ and $\mathrm{h}$ then provide the desired canonical morphism $$ \mathrm{h}\left(\mathcal{D}(\mathsf{A})\right)\to\mathsf{D}(\mathsf{A}). $$

We immediately see that this morphism is an equivalence, since $\mathrm{h}$ being a left adjoint implies it commutes with colimits, and $\mathrm{h}\mathrm{N}(\mathsf{C})\simeq\mathsf{C}$ (naturally!) for all $\mathsf{C}\in\mathcal{C}\mathsf{at}_1.$

Additionally, we obtain the desired result for $\mathcal{D}_{\mathrm{qc}}(X)$ (despite the fact that this is not the derived $\infty$-category of some abelian category in general) by noticing that the category $\mathcal{D}_{\mathrm{qc}}(X)\subseteq\mathcal{D}(X)$ is defined by a condition on cohomology (which $\mathrm{h}$ respects).

In fact this construction then provides a functor from ``relative $\infty$-categories'' (i.e., the $\infty$-category whose objects are pairs $(\mathcal{C},\mathcal{W}),$ where $\mathcal{C}$ is an $\infty$-category and $\mathcal{W}$ is a class of morphisms in $\mathcal{C},$ and whose morphisms $F : (\mathcal{C},\mathcal{W})\to(\mathcal{C}',\mathcal{W}')$ are functors of $\infty$-categories $F : \mathcal{C}\to\mathcal{C}'$ such that $F(\mathcal{W})\subseteq\mathcal{W}'$) to the $\infty$-category of $\infty$-categories.

The desired property that these functorial equivalences should be compatible with derived pullback of sheaves should follow from expressing the derived pullback as an appropriate functor of relative $\infty$-categories, and noting that there is functoriality of these pushouts, but I have to think about this aspect of the construction more.

If I've made any errors in my recollection of the argument I learned, or if there are technicalities or subtleties I'm not doing justice, please do let me know. I'll leave the question open for now since the question of compatibility with derived pullbacks hasn't been completely answered yet.