That's only true for systems where the principle of explosion actually holds though, isn't it? So it wouldn't apply to paraconsistent systems.

In the end, Gödel is actually giving us a choice: Either accept incompleteness or accept inconsistency. Of course it's true that historically incompleteness has been perceived as the only viable choice, but at least a few paraconsistent logicians like Graham Priest have argued for (non-explosive) inconsistency instead.