I used to think this scenario was likely, so do many prepper types. Now that I am older and started delving into my own ethnic history I no longer believe your scenario is likely apart from a short term panic reaction.
My grandparents lived thru scenarios way worse than you have described and the exact opposite happened, in fact most of history refutes your scenario.
Folks instead tend to become more moral, more generous and more humane than they currently are now. Religion and tradition is the glue that circumvents the disaster you describe.
But still the answers is smatchet just for good measure.
Yea , I don't know . I couldnt have ever imagined seriously considering it myself . The internet and social media more importantly has tactically divided this country in a way not possible before using weaponized connections and targeted marketing and ads to make money . Netflix of all places just dropped a documentary on it recently called " The Social Dilema " About how google facebook and twitter are manipulating us into civil war .
So I dont think these specific dynamics of this current paradigm have ever been seen in history prior to this time period. And people are virulently polarized and you can see the pot simmering . So now ? I'm not so convinced we won't see an instance of the event horizon I outlined. And I'd rather be ready for it than not and hope it doesnt happen .
To your suggestion , you dont think a smatchet is too small ? My train of thought is if I was facing a small patrol sized squad Id want something a little longer to keep them at bay . Smatchet distance they have the option to start grappling to disarm you . And you cant get much of a swing on multiple tangos at that range.