Very little of research in programming languages is anything like “We tried it out on a bunch of developers and the data support our conclusion that ...”.
PL is not an empirical science in that sense, it's more like mathematics. So your initial intuition about type safety, etc. is right on the money—that's the academic side of it.
In bob's setting, the computation system is not determined or fixed. It is open-ended...
In intuitionistic mathematics, we of course accept that there is no Turing computable halting oracle, but we do not rule out the possibility that there is some other effective oracle that can decide halting. (This is in contrast to recursive mathematics / Russian constructivism)
Bob is coming from the perspective of Brouwerian Intuitionism, and so he is not really that interested in a closed "formal system". So, "proofs" in the setting that Bob cares about are not derivations in a formal system, so Gödel's result doesn't really apply.
PL is not an empirical science in that sense, it's more like mathematics. So your initial intuition about type safety, etc. is right on the money—that's the academic side of it.