It’s because “attack” isn’t specific enough. Everything in DnD is either a weapon attack (typically a physical attack using whatever weapon you have equipped) or a spell attack. In general parlance, “I punch the kobold” translates to “I use Unarmed Strike to make a weapon attack on the kobold.” But that doesn’t mean the Unarmed Strike is a weapon. Since generic attacks aren’t allowed in the rules, you have to designate it as a weapon attack, instead of a spell attack. Oftentimes, the distinction is because there are certain spells or effects that use your weapon as a spell focus, or trigger when making/taking weapon/spell attacks.

For instance, Booming Blade requires brandishing a weapon to channel the spell before you make a weapon attack. The spell component literally lists “a melee weapon worth at least 1 sp, which the spell does not consume.” Then if you hit with the weapon attack, the spell triggers. So your fists could make a weapon attack (using Unarmed Strike) but would not count as a valid weapon for the spell. Even if you could convince the DM that your hand is worth at least 1 silver piece, it still wouldn’t be a melee weapon. So you wouldn’t be able to cast the spell if you were unarmed.

Maybe. It's because "weapon attack" is the verbiage they settled on for hitting somebody with something that isn't a spell (spells make "spell attacks"). They could call them "weapon or unarmed attacks" but that seems unnecessarily verbose when 95% of them are going to be made with a weapon. You might think that for hand-to-hand combat you could simply refer to "melee attacks," but "melee" is a specifier that can be applied to spell attacks too, so it's out.

So the current situation is this: a rule can simply refer to all "attacks," or it can refer to "melee" or "ranged" attacks, or it can refer to "weapon" or "spell" attacks, or it can use both specifiers (as in "ranged weapon attack").

So if you want to fix it, you need a word to replace "weapon" that could include unarmed combat but excludes all spells. "Physical" might be good, but has some edge case problems: if I have a psychic "blade" that attacks your mind, it makes "physical attacks" despite being a non-physical object. If I have a spell that physically throws a boulder at you, it's pretty easy for me to remember that I should make a spell attack roll, but if you have a feature that defends against "physical attacks" you might think it should apply against the boulder when it doesn't. "Martial attack" might be getting at the right thing, but it sounds strange, and for new players who might be new to RPGs "martial" and "melee" are both uncommon words that kind of sound similar, and that might cause confusion. (Also "martial melee attack" sounds more natural than "melee martial attack," but then it has the opposite word order from "melee spell attack" and that's weird.)

There may be a perfect word out there, but in the end they decided "weapon" was the least confusing, despite requiring the caveat that attacking unarmed is a "weapon attack." And so everywhere that the rules say "attack with a weapon" instead, it is to specifically exclude unarmed attacks, although I admit that it's not always obvious why they want to do that.