I love this post and not just because I appreciate realism in gaming.

When I was in second grade a girl I barely knew and a boy I only slightly knew more were wrestling. I thought he was about to lose the upper hand so I intervened and the girl and I wound up rolling down a small hill, I don’t recall if that was part of my attack plan but let’s just say it was.

Now at the bottom of the hill she wound up with the high ground and while I lay on my back with my stomach exposed. She took this opportunity to bite me and while similarly it didn’t break skin I was still taken to the hospital for shots. Cassie, I still think about you sometimes.

Given the nastiness of the human mouth, "fight bite" (hand cuts from accidentally punching teeth) is often treated with antibiotics. On that basis I'd stick with at most 1d2 damage, plus lingering status effects to the hand unless you have disease curing abilities or medicaments. Maybe the damage sticks unless specially treated, and untreated you can lose the hand to infection?

Yeah, a bite is more like a dagger than a short sword. Plus barbarians already get an extra +2 damage while raging, which makes their bites average more than a d6 plus strength with the same max damage.

Beast barbarians can even cause their teeth to grow during a rage giving them a d8 bite attack.

I mean, if you're going to attempt to kill with a bite, I'd assume they're going for the neck or other particularly vulnerable place. I'd argue commoner bite attacks should have very low chance to hit, but could plausibly be lethal. Most places they get a good bite won't be able to do any damage, except maybe cause a Con save for disease.... I'd say an improvised attack with Disadvantage for 1d6 damage probably works pretty well.