I think I need a slight clarification, because I was promised this works for every one relator group, regardless of the relator. I know the Freiheitssatz extends to the case when the relator does not necessarily involve all generators, but the chosen subset still excludes a generator involved in $r$. However I'm not so sure this proof extends in a similar way. Take the case where $r = g_1^k$ involves a single generator, then it seems to me that any HNN extension (at least, the construction used here) would necessarily have at least three generators. Is there something I'm missing here?

No. It is definitely an HNN extension of a free group. I just posed the more general question because I didn't know of this theorem on HNN extensions, and thought to answer it from a different angle.

This is unfortunate for at least one author who I'll leave anonymous, because it provides a counterexample to a theorem in one of his books on one relator groups.

I apologize for the ignorance, but is that a variation/application of Helly's theorem on convex sets? Could you point me to a proof of the sufficient condition above?

This is the proof that was communicated orally to me, but I was looking for the name or to see it in a book somewhere. This is just what I needed. Thanks!