Read the MM. Claws, slams, tentacles, bites, rakes etc are also melee attacks and melee weapon attacks, but not weapons.
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See the difference?
I am not arguing that there is a failure of an actual distinction to be made; nor that there is a failure of the mechanics--though there are still flaws with your argument (see below).
I'm arguing that
the terminology is confusing as hell due to basic logic. People expect properties to be transitive unless there's a good reason for them not to be. (A+B) + C = A + (B+C). If something is a "melee attack," and also a "melee weapon attack," then applying Natural Language and intuitive logic, it should be a weapon. But it is not: it has property 1 (melee attack), which is described "A," and property 2 (melee
weapon attack), which is described as "A+B," but
does not have property 3 (weapon), which is described "B." This unexpected failure of transitivity is confusing until carefully explained. Thus, either the term "melee weapon attack," or the decision to make unarmed strikes (and similar "natural offensive" things) not weapons, is a mistake
for terms of comprehension.
Unless you can somehow demonstrate that people shouldn't intuitively expect such transitivity to hold, I don't see how you can get around this issue of "it's a melee attack, and a melee weapon attack, but not a weapon."
Personally, I would argue that the term should be melee
physical attack, as opposed to a melee
magical attack: physical attacks being those which are not the result of manifested magical forces, but rather through a part of a creature's body or an item (such as a weapon) held, thrown, or loosed by part of a creature's body. If a spell causes a physical object to come into existence, then any such spell will specify whether the attack is magical or physical in nature. So
flame blade, which summons magical energy in a space essentially "as though" it were a weapon but is not actually a physical object, would use a melee magical attack.
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You argue that these things are not weapons, but
alter self allows for the addition of
natural weapons...which specifically affect your "unarmed strike." So now we have a clear-cut example of something that is even
called a "weapon," yet it is still improving unarmed strike damage--and the description of the spell specifically refers to "claws, fangs, spines, horns, or a different natural weapon of your choice," which includes things you yourself referred to as
not being "weapons."
Unless we're now taking the even more nonsensical tack that "natural weapons" are not "weapons."