Are the Haag axioms partly motivated by the fact that in the general case, it is impossible to form a differential algebra with (basic) Schwartz distributions? I got the impression due to the following fact :
In constructions of AQFTs, the whole algebra seems to be defined from the field operator, and not the canonical momentum. While not all QFTs are defined from a Lagrangian, it seems odd that no one would at least try to include it in some variation. On the other hand, $[\varphi(f), \pi(g)] = i \langle f, g \rangle$, meaning that $\delta(x-y)$ would be part of the algebra if we included it, which responds famously poorly to the product $\delta^2$.
Instead the field operator is defined directly via its propagator when the algebra is constructed, which seems like a hard thing to show if we did not know the theory already.
Also any quantity implying components of the stress energy tensor, which might cause the same problems, are not part of the algebra either but just part of the Poincaré isomorphisms of the algebra.
Are those choices partly motivated by the fact that distributions do not multiply trivially or am I mistaken?