Except that if it was special, with it's own subset of rule, it would have to actually have it's own subset of rules. If you're right, and it is a natural weapon, then there is nothing anywhere which tells you what these special rules are, which means that either:
1. It isn't a natural weapon.
2. It follows all the same rules as other natural weapons.
3. You have to make stuff up.
Except that we have the section of the combat rules and tables that show us explicitly how unarmed strikes work, and we have ALL noted how they differ from both other natural weapons and manufactured weapons. To use an old, old phrase, they "are neither fish nor fowl."
As for us "making stuff up," we would counter that you are completely ignoring explicit text in the PHB. You SAY that the spells and other sections are wrong, but you don't have any explicit text to counter it. In other words, we would say your list should ACTUALLY read:
1. It isn't a natural weapon despite sections of the Core rules that explicitly call it a natural weapon.
2. It follows all the same rules as other natural weapons.
3. You infer that the designers had intended that unarmed strikes be considered a subset of natural weapons, but neglected to say so explicitly.
How does Occam's Razor cut?
First, 2 is clearly wrong- both sides have established that unarmed strikes have their own rules.
If 1 is correct, we would have to assume that the sections are wrong with only inferred (not direct) evidence. If the statement "unarmed strikes are natural weapons" is an error, it is one that has not only not been caught, but has been repeated. The fact that these sections some would excise made it through an initial printing of the core rules, a revision of the rules, and into subsequent releases of supporting material by this company,
all while under the eyes of multiple editors, is an indicator that it is NOT an error.
If 3 is correct, the designers wanted unarmed strikes to be considered a subcategory of natural weapons with special rules. It is, as we have demonstrated, an inferrence supported by numerous explicit references to unarmed strikes as natural weapons in the Core rules.
I'm thinking Occam's Razor cuts in favor of 3. Instead of inferring that several sections of explicit text are simply wrong, position 3 inferrs that
as little as 1 sentence was accidentally omitted, namely a stand-alone sentence that says "Unarmed Strikes are a subcategory of Natural Weapons" (like in the glossary, as opposed to in a spell description).
After all, RAW, there are NUMEROUS references that equate unarmed strikes with natural weapons, and we ARE (or were at some point) discussing this as a question of RAW.