This paper provides an alternative implementation of the principle of general local covariance for algebraic quantum field theories (AQFTs) which is more flexible and powerful than the original one by Brunetti, Fredenhagen and Verch. This is realized by considering the $2$-functor $\mathsf{HK} : \mathbf{Loc}^\mathrm{op} \to \mathbf{CAT}$ which assigns to each Lorentzian manifold $M$ the category $\mathsf{HK}(M)$ of Haag-Kastler-style AQFTs over $M$ and to each embedding $f:M\to N$ a pullback functor $f^\ast = \mathsf{HK}(f) : \mathsf{HK}(N) \to \mathsf{HK}(M)$ restricting theories from $N$ to $M$. Locally covariant AQFTs are recovered as the points of the $2$-functor $\mathsf{HK}$. The main advantages of this new perspective are: 1.) It leads to technical simplifications, in particular with regard to the time-slice axiom, since global problems on $\mathbf{Loc}$ become families of simpler local problems on individual Lorentzian manifolds. 2.) Some aspects of the Haag-Kastler framework which previously got lost in locally covariant AQFT, such as a relative compactness condition on the open subsets in a Lorentzian manifold $M$, are reintroduced. 3.) It provides a successful and radically new perspective on descent conditions in AQFT, i.e. local-to-global conditions which allow one to recover a global AQFT on a Lorentzian manifold $M$ from its local data in an open cover $\{U_i \subseteq M\}$.