On the non-reflective cases we at least see that we can forge small but very hard problems too. The most simple ones are the infinite loops testing some whole number related property, like: Goldbach conjecture, Fermat's theorem.
Now we know the program of the second case never halts but it was not an easy proof, the first one is still open.
But in fact, the whole proof is based on the self-application problem, where it's roughly analogous to the liars paradox. But the proof doesn't say anything about other, non-"reflective" cases. Theoretically, it might be possible to build a program that makes correct decisions whenever that's possible, and otherwise ends up in an infinite loop (e.g. in the self-application case). I wonder if there are any efforts in this direction; so far I've never heard of it.