If you need to analyse the English language to this degree to ensure no-one is confused by unarmed strikes not being a weapon, chances are there's something wonky going on. Barring a compelling reason, I don't see the benefit in forcing people to have to go to such an in-depth analysis and would instead say melee weapon attacks are in fact weapons. See how much easier that is?
I went to that depth only because, had I not done so, I would have said little beyond, "Those aren't verbs, they're nouns," which would have invited a "you're just bickering" or "says who?" retort. I was heading that off at the pass by being thorough, and as I said at the start of the post, it had nothing really to do with my position on whether "unarmed strike" being an "attack," and a "weapon attack," but
not a "weapon" is confusing.
There are really two simple approaches, if you feel a change is necessary. One: As you have done, allow that if something is referred to as a weapon in a specific/modified sense (natural
weapon, melee
weapon attack), it is a weapon. This has the advantage of requiring effectively no real change, other than broadening the reference of the term "weapon," but has the disadvantage of potentially allowing unforeseen, and unwanted, synergy now or in the future. The other: Eliminate the terms "melee
weapon attack" and "natural
weapon," as they are confusing (because they include, to varying degree, things that aren't weapons), and replace them with terms that do not invite such confusion, like "melee
physical attack" or "melee
object attack"/"natural
offense(s)" or "natural
armaments." This has the advantage of restoring the natural-ness of the language used and avoiding unwanted synergies (potential or actual), but requires post-release jargon changes and the introduction of perhaps-overly-technical-sounding terms.
(There is a third option but it's probably beyond the pale for most people: stop using the word "weapon" to refer solely to manufactured implements of violence, and instead define those things to be "arms" or the like, and having "weapon" be a higher order abstraction; then "melee weapon attack" would become "melee armament attack," and certain non-armament weapons would be allowed to qualify as a "melee armament attack" even though they are not, strictly speaking, "armaments." But this is so highly technical-sounding as to be possibly worse than the current situation, so I don't really think it's a valid option.)
The whole surprise attack rules also seem equally perplexing, but there is a (supposed, I haven't seen anyone math it out) balance reason there so I would say that is a compelling reason. Unarmed strikes seem to be lacking that compelling reason.
I think "preventing accidental synergies" is a valid reason to want to silo things into their own categories. I just strongly dislike the tortured logic required to make sense of the mandated synergies, when 5e
supposedly values "natural language" over precise terms.