An unarmed strike is, by definition, not striking with a weapon. Trying to twist that around is going to give you a headache. A goal of the 5e rules design is to be able to guess at the rule and be right most of the time.
If you didn't read the rules at all and were just asked:
"Can a Monk use Stunning Strike with Unarmed Strike?"
and
"Can a Kensai choose Unarmed Strike as their special weapon?"
What would your answers be?
I think the vast majority of people would guess the correct answers.
Now, could they have described the difference between weapon and spell attack differently? Yes. But, when the distinction is relevant it should be easy enough to guess at anyway.
Is there any reason for this perverse "rule"? Does it really imbalance anything if unarmed strikes are just considered another melee weapon, treated like all others?
I can think of 2 reasons.
1. Theme. It's weird to treat the body as an object which can be enchanted, etc.
2. Rules are supposed to be simple enough that they can be guessed at. A rule that makes the character's body count as a separate weapon would be confusing.