In Quantum Field Theory one often needs to compute the $n$-point correlation functions $\omega_n(x_1,\dots,x_n)$ which are traditionally given as vacuum expectation values
$$\omega_n(x_1,\dots,x_n) = \langle \Omega | T\{\phi(x_1)\cdots\phi(x_n)\}|\Omega\rangle.$$
In order to do this there are, as far as I know, two main approaches with two main issues which actually invalidate these two approaches mathematically:
The first approach is work out a relation between the interacting fields $\phi(x)$, the vacuum $|\Omega\rangle$ and their free theory counterparts $\phi_0(x)$ and $|0\rangle$. This is achieved by switching to the interaction picture. The issue here is: Haag's theorem states that this cannot be done - the unitary transformation that leads to the interaction picture doesn't exist.
The second approach is to derive the Schwinger-Dyson equation. This is done in Matthew Schwartz book. This doesn't use the interaction picture, but still makes a false assumption: it assumes the canonical commutation relations are obeyed for the interacting fields, which as far as I know isn't true.
Now, Algebraic Quantum Field Theory seems a nice approach to QFT. In this article Robert Wald gives an introduction to the topic, and it seems at first that those problems aren't there at all.
My question here is: concerning the issue with the interaction picture and Haag's theorem, or the Schwinger-Dyson equation counterpart based on the canonical commutation relation, Algebraic Quantum Field Theory gives any workaround to deal with $n$-point functions rigorously?
Does AQFT allow to relate the $n$-point functions to the perturbative expansion in terms of the free fields pictured as Feynman diagrams in a rigorous way? Or it gives an entirely different way to compute the $n$-point functions?
What AQFT actually solves in this matter?