Yes, I'm aware what the rules say. And those rules specify that an unarmed attack is one option when doing a melee attack. And there are other rules that specify when you can make a melee attack. OPs post was noting the weirdness of D&D, in that there are some things that aren't explicitly specified in the rules. Specifically, whether using two fists counts as dual-wielding (RAW, it doesn't).

According to the rules, characters can make a melee attack when performing the Attack action (plus in a number of other cases). Most of the time, the Attack action involves one or more attacks with a weapon (martial classes get more than one starting at level 5).

So any weapon attack can be substituted as an unarmed attack. A character wielding a greataxe who can normally make two attacks with the Attack action could substitute one or both of those attacks with kicks, elbows, or for flavor, releasing the weapon with one hand and bitchslapping their opponent.

The question isn't whether someone wielding other weapons can make an unarmed attack, it's a question of when. More specifically, when can a character use a bonus action to make an unarmed attack.

The rules also contain information about dual-wielding weapons:

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.

OP's post calls out that fighting bare-fisted would not qualify as two-weapon fighting, and thus RAW a character fighting unarmed could not use a Bonus Action to make an additional attack (despite "wielding" two fists).

My point was that, as a GM, I would rule that fighting unarmed, or fighting with a single one-handed weapon and not having a shield, would qualify as being able to make an additional attack with a bonus action per the two weapon fighting rules.

But per the rules, landing an unarmed attack in this scenario would result in a maximum of one (1) point of damage, as the Str modifier would not be added to the damage (unless the character had some other benefit that improved it, such as a class feature or feat). So there's no reason to not allow it, as it's a pretty weak option.