Natural weapons are not weapons - therefore they are not simple weapons.
Where is this stated? The closest I can see is in the SRD (p.262), and that suggests that natural weapons are weapons ("These can be spell attacks or weapon attacks, where the 'weapon' might be a manufactured item or a natural weapon, such as a claw or tail spike.") That 'weapon' is put in quotes is not enough evidence for me to conclude that natural weapons aren't weapons, given the context. After all, a manufactured item can count as an improvised weapon, and most players (though not the rules) would not consider an improvised weapon to be equivalent to a 'real' weapon. (See, now I'm doing it!)
Unarmed strikes are not weapons, that much is clear (and the weapon list was specifically changed to remove that confusion). The only other reference I can find is under the spell description of Alter Self, but that explanation doesn't help, because that spell equates natural weapons and unarmed strikes as if they are interchangeable. ("You grow claws, fangs, spines, horns, or a different natural weapon of your choice. Your unarmed strikes deal 1d6 bludgeoning, piercing, or slashing damage as appropriate to the natural weapon you chose..." SRD 5.0, p.115). I suspect the spell's wording is deliberate to allow a monk to cast the spell on herself and continue to use her Martial Arts with her unarmed strikes, though higher level monks would have little reason to cast the spell.
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Pauper