Wait. Fists Are Different Than Fists?

Unarmed Fighting

Replace the following 3rd paragraph under Combat/Melee Attacks on p195 of the Players Handbook with:

While one or more of your hands are empty, instead of using a held weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow. You are proficient with your unarmed strikes. Your unarmed strikes are considered to be simple weapons with the light and finesse traits. On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength or Dexterity modifier. These weapons are considered both to be parts of your body and 'held in your hand' as appropriate for purposes of rules that reference handheld weapons. They are never considered to 'occupy your hands' for the purposes of rules interactions that require your hands to be empty, nor do they count as an object. Your unarmed strikes cannot be disarmed by any means and do not have any value for the purposes of other rules interactions. You cannot use an unarmed strike during an attack action which utilizes the rules for Two-Weapon Fighting.

Natural Weapons

Apply the following changes to the many disparate places where they otherwise crop up:

Features of certain races and classes allow them to manifest or use natural weapons, such as claws, teeth, or weaponized appendages like tails or tentacles. In all circumstances, these will be considered simple weapons with the light and finesse properties, unless otherwise specified.
These weapons are considered both to be parts of your body and 'held in your hand' as appropriate for purposes of rules that reference handheld weapons. They are never considered to 'occupy your hands' for the purposes of rules interactions that require your hands to be empty, nor do they count as an object. Your natural weapons cannot be disarmed by any means and do not have any value for the purposes of other rules interactions. You cannot use a natural weapon during an attack action which utilizes the rules for Two-Weapon Fighting if you have a shield equipped.

It seems like every source of natural weapons interacts differently than others, the language is simply not unified, and should be, plus it is unrealistically penalized for the sake of other rules interactions: This affects the Blood Hunter - Order of the Lycan’s Claws, a Monk's punches and kicks, a Path of the Beast Barbarian's natural weapons, and many races’ innate claw, horn and bite attacks. This unifies the rules somewhat, eliminating most problematic rules specificity and shoring up certain weaknesses of using them, and applies a little bit of logic
Whether your teeth or claws are racial, magical, class-based, etc. They should all be treated the same and interact with the rules in a similar fashion.