No, I get your point, and while it is confusing a bit, I don't think it's as confusing as you're making it out to be.
As someone stated, there are only 4 categories, so Unarmed strikes have to fall into one of them. They fall into melee weapon attacks because that is the only bucket they can realistically fall into (not being ranged or spells), but they are not in fact "weapons"
From the Sage Advice Compendium:
The game often makes exceptions to general rules, and this is an important exception: that unarmed strikes count as melee weapon attacks despite not being weapons.
So any time you see "weapon" or "weapon attack", Unarmed strikes don't work. Any time you see "melee weapon attack" you can use unarmed strikes just fine.
If what you want is a melee brawler that is NOT a monk or someone without the Tavern Brawler feat... you're going to have to house rule or home brew. That is specifically the Monk's wheelhouse and right now RAW they don't want others treading on it. Let the fighters fight and the monks brawl with unarmed strikes. It's what they're best at.
You could probably get away with a Barbarian (Berserker) as I noted above, though without the Tavern Brawler feat you're limited in what damage you can deal.
Heck, now I want to do a Barbarian (Berserker)/Monk multiclass with the Grappler feat, a more controlled Monk/Fighter (Battlemaster), or Monk/Paladin for those unarmed smites.
Those actually sound fun, or some combination of them
Some of those fun combinations is what got me in this spot!

The Frenzy Berserker was one of the first that got me wondering about it all. Ideally, even if it is only a couple ways, there
should be ways to make a brawler without having to dip into monk. Why an official Rogue subclass "Thug" has never been made is beyond my understanding.
Anyway, I definitely planned to take the Tavern Brawler feat (level 1 as human variant). To me allowing anything that could apply to attacks with a weapon apply to unarmed strikes won't disrupt game balance or take away from monks. Since monks unarmed damage improves, I am not concerned with another class pugilist stepping on their toes.
So, I have to disagree with the SAC though because some of it doesn't make any sense to not include unarmed strikes
as weapons. Something like Horde Breaker says "weapon attack". I think the reason it says weapon attack was so people understood it could be used with
either a melee weapon or a ranged weapon, but not a spell. To me this was a simplification so when a feature was applicable to any form of weapon attack, they didn't have to write "melee or ranged weapon" each time.
I'll discuss it with our DM and table, but we will probably stick with the original unarmed strikes rules. For one thing, not everyone is proficient with unarmed strikes--a lot of people can't fight to save their skin. I like keeping them on the weapon table under Simple Weapons. The only classes who would not be proficiency would be Sorcerers and Wizards. This also allows Tavern Brawler to keep the "you are proficient in unarmed strikes" feature. At the very least, we will probably augment Tavern Brawler so your unarmed strikes
are now considered weapons. Monks will also gain that feature. This way a player who wants a brawler can take Tavern Brawler and use TWF to gain a second attack with the bonus action if they want. And just to note when Monks use the bonus action for an unarmed strike, they
do get to add STR or DEX to damage. Other characters won't unless they used the Fighter Two-Weapon Style.
I just watched
Gladiator this weekend as well. While the movie isn't "great", the combat and fights in the arena show how (in most cinema and in historically as well) combatants using weapons often made unarmed strikes as part of their routine. In 5E there is no reason why anyone would normally do this as weapons deal more damage (in general) than unarmed strikes. But then the game doesn't reflect what we commonly see on screen and also how people fought historically.
This revision for unarmed strikes IMO only made things worse and more confusing. When unarmed strikes were listed under simple weapons, they were weapons and it was simpler. (No pun intended...)
I would not question your time playing, but the age of the copy of the PHB you bought. This was official errata from 4 years ago, so it would appear this way in probably any printing of the PHB after the 2nd or 3rd printing So if you bought your copy new a year ago, it should have been at least a 5th or 6th printing. If you really have that old a copy, you may want to check the rest of the errata against your book.
Actually, I bought two copies online about a month apart, the second for when new players show up or for the guys who don't own a copy. Here is the strange thing, one copy
is older and has the original rules and the other is obviously newer because it has the revised rules. By chance, I've been using the older one and didn't even think about it. In fact, I never realized they were different versions until all this. I guess when I bought the older copy the store simply had it on their shelves for a while...