completely formalized.
Our starting point is similar to yours, so to answer your question, I
would say that yes, that idea can be made to work. However, at least in
our approach, most of the work is in proving that the Hurewicz map is
actually a group homomorphism. In that argument we really need the fact
that homology is defined as a colimit and we can't just use some
sufficiently large t.
Another key ingredient in the approach is the relation between smash
products of types and tensor products of their homotopy groups, which
we had to develop for this purpose.
For completeness, note that Floris van Doorn points out in his thesis
that another possible approach is to use the Serre spectral sequence for
homology. In this approach one still has to prove the dimension 1 case.
Best,
Luis