At the level of IR-finite, individual Feynman graphs, indeed EG is just as mathematically rigorous a renormalization scheme as the more usual BPH and BPHZ schemes. We are still doing formal perturbation theory (i.e. formal power series in the coupling constant) in either case. The reason why EG is associated with algebraic QFT is a matter of concept rather than rigor.
The main characteristic of the EG scheme is that it is formulated in position space, rather than in momentum space as BPH and BPHZ are. This allows a clear separation of the IR problem from the UV problem by means of adiabatic switching of the interaction by a smooth cutoff function with compact support in space-time. Doing this in BPH and BPHZ is very messy. This separation is important because the UV problem concerns the local observables of the theory, whereas the IR problem concerns the latter´s state space - the solution of the UV problem is IR-cutoff-independent, and the EG framework makes this very precise.
The separation of the UV problem from the IR problem by means of adiabatic switching allows one to define a net of local observables for each order of perturbation theory, by means of families of Bogolyubov´s S-matrices parametrized by all cutoff functions - the definition is such that the local algebras become cutoff-independent. It is in this sense that EG is nowadays more commonly associated to algebraic QFT - in fact, the branch of AQFT dealing with the perturbative construction of nets of local observables is usually called perturbative algebraic QFT since we are still dealing with formal power series in the coupling constant, and the EG scheme is simply the one that fits most naturally into the philosophy of (P)AQFT. One can go even further and write the EG scheme no longer in terms of individual graphs, but as a renormalization scheme for a formal deformation of the algebraic structure of the free net by the interaction.
At each order of perturbation theory, the net of observables can be shown to obey the Haag-Kastler axioms, apart from the vacuum axiom. The latter, of course, does not hold in an unmodified form since the cutoff interaction is not translation invariant, but this again is a property of the state space of the theory. Solving the IR problem amounts to building it for each order of perturbation theory by carefully removing the cutoff. Of course, here the devil is in the details, and the situation is no better than in standard QFT. In any case, the IR-cutoff theory is well-defined as formal power series for each choice of cutoff function.