I'm not saying that reality is encompassed in the theorem because "mathematics", I'm positing that if mathematics, a man-made formal system, is self-consistent but requires a true, unprovable statement, then perhaps other man-made systems might reveal similar paradoxes, when put under scrutiny.

Mathematics is unique in that it is systematic and, as Godel discovered, able to be self referential and explicitly outline this paradox. Yet one can recreate the sense of the theorem in English with the phrase "this sentence is false".

This doesn't invalidate mathematics, nor English, nor any system. It simply demonstrates that what we may consider the "sound" logic of mathematics is paridoxical, and I believe that this paradox, this strange loop, is not unique to Godel's mathematical theory alone.

Philosophers don't (or don't all) use Godel's theorem to _prove_ something beyond that in an axiomatic way. They don't extrapolate, they use it as a raw information / material for further thinking.

In that role, the limited conditions under which the proof is true doesn't really matter. Only the fact that under such limited conditions, this (Godel's) result can occur matters -- which is philosophical far from evident, Hilbert/Russel etc thought they could axiomatize math without such contradictions.