Firstly, I think you mean the Wightman axioms, not the Haag-Kastler axioms, because only the Wightman axioms include an operator-valued distribution such as $\hat\phi(x)$.
Certainly the immediate introduction of the propagator is somewhat abstract, but part of Wightman's intention at the time ---almost 30 years after the first attempts to construct quantum fields, so it's not like it was hasty--- was perhaps to introduce a clearer, more abstract definition of what a quantum field is, instead of using only the rather pragmatic constructions of working Physicists, so that there could be a more solid basis for mathematicians to consider the theory. The construction of the momentum operator $\hat\pi(x)$, given $\hat\phi(x)$ at all points of space-time, and of the consequent commutation relation, is close to immediate.
Though it can be subliminal, the question of how to handle the distributional character of Poincaré invariant propagators on the light-cone in real space, and, rather associated and as significant, how to manage the scaling properties near the light-cone, permeate discussions of quantum fields. So yes, various aspects of distributions are significant.
It is important in many of the proofs that start from the Wightman axioms that the restriction to positive frequency introduces a many-variable complex analyticity that is rather analogous to the analytic property of the Hilbert transform. This property makes the multiplication of distributions sometimes possible. Note also that the convolution of distributions is often possible.
The Wightman axioms did allow a lot to be clarified about quantum field theory. Various theorems were proved. People also tried to weaken the axioms in various ways, which can be seen discussed, for example, in Section 3 of R.F.Streater, Rep. Prog. Phys. (1975) 38 771-846, "Outline of axiomatic relativistic quantum field theory". Mathematicians and Philosophers are often more likely to work with Local Quantum Physics, one of the modern names for constructions that start from the Haag-Kastler axioms, for which it's probably best to start from Rudolph Haag's book, "Local Quantum Physics" (which is certainly also a great source for a discussion of the Wightman axioms).