Obviously Australian culture has not come out of nowhere, but a simple line from its 'convict' past cannot be drawn as anything other than a very poor caricature.
Whereas there are of course historical/cultural contributors to Australia's sanity re arms (notably: a liking of order, and tolerance of authority where it can help maintain that order), there are many others more contemporary in origin. We don't have a large arms industry (our governments choose instead to be corrupted by fossil fuel corporations; arguably just as dangerous). We have had genuinely courageous leadership at pivotal moments. We are not burdened with an archaic constitution. We tend to dissimulate regarding our violent history rather than valorise it. All of these features have their negative contributions in some policy arenas, but have helped nudge us away from a strong gun culture (which has at moments had the potential to emerge).
This has very little to do with "us" having been "sent here".