Really? I have never allowed that unless there is a specific exception that makes the hands count as a natural weapon, in any version of D&D.
Why?
Sure, I'll grant you that we didn't always allow it previously, but that's mainly because nobody ever did it because nobody was proficient with unarmed strikes in previous editions and that was a significant penalty by itself through 3e.
However, let's look at it from a game design perspective and not a "read the book back to me" perspective. Why might we want to make unarmed strikes not count as weapons?
From a narrative sense, does it make sense to smite a punch? Oh yeah. Media is full of super dramatic punches and martial arts backed by extraordinary powers. Is it cool to imagine a Paladin PC smite-punching a vampire or demon and killing it? Oh yeah. Magic to enchant someone's fists and make them capable of mighty deeds? Oh yeah, that's cool. Narratively, treating a fist like it's a weapon has some cool benefits.
Also, since all you need to arm yourself is a knife or a club and suddenly everything is a go, you're not really restricting that many situations in actual play. Like it just doesn't come up in every campaign that you're an unarmed character and you need to fight to defeat your opponents.
From a game balance sense, what's broken about it? An unarmed strike is a 1 damage weapon for most characters. Do we care if people can optimize them? Are we concerned about Monk/Paladins burning Ki
and spell slots to get one extra attack and one extra smite over TWF or Polearm Master? Burning the candle at both ends on resources you have half as many of in classes that require, what,
four good stats to pull off really well? Martial Arts doesn't work while wearing armor, so you need Dex and Wis. You'll be in melee, so you need Con. Paladin abilities key off Cha. And both Paladin and Monk scale well single class, Paladin gaining more spells and Extra Attack plus unique defensive abilities powered by Cha, while Monk gains more Ki to fuel Martial Arts, scaling Unarmed Strike weapon die, and abilities powered by Ki, Dex, and Wis. You're giving up a ton of potency in order to burn through your nova faster, but you're worse at nova because you've got fewer resources, too.
So is it single class Monk? We're worried at 11th level the Monk can get 4 attacks at 1d8+Dex+2d4 (14.5) with Elemental Weapon instead of... 2 attacks at 1d8+Dex+2d4 (14.5) and 2 attacks at 1d8+Dex (9.5)? That's what we're upset by? The fact that Elemental Weapon as a 5th level spell affects all 4 attacks and not just 2? The extra 4d4 is the breaking point? Well, why does an 11th level Fighter get 4 attacks with one weapon with Polearm Master or Crossbow Expert then (and then with only two good stats)? Or are we worried about 17th level Paladins casting
holy weapon or whatever it is on their hands (which I still say is pretty freaking cool)?
Are we worried that a Paladin player is going to think his 1 damage fist is a better choice than a 1d3 club, 1d4 dagger, 1d6 short sword, 1d8 longsword, etc.? Do we think players are that bad at math? They want an off-hand fist instead of a dagger because... what, their DM won't let them drop their off-hand weapon to cast a spell?
If we're worried about all that, why aren't we worried about Tabaxi, which have a bonus to Dex and Cha and natural weapons? Aarakocra? Leonin? Lizardfolk? Tortles? Boy they do like giving races natural weapons, don't they?
So we're not really
stopping an exploit.
So:
a) It tells a good story
b) It's cool
c) It doesn't come up
that often
d) If you try to optimize for it, you kind of can't
e) Even if you ban it, you can still do the optimization for it with some player races, and it's still not that great
f) And even if there is something that is overpowered, maybe the
real problem is that we shouldn't let Paladins have smite more than once or twice a round, or the same target more than once a round? Or use a different kind of situational limitation like Sneak Attack has?
So... why ban it? The game benefits by having less restrictive rules. That was one of the foundations of 5e. The fewer the rules that restrict options, the less rigid and more flexible the game is. The fewer the rules that restrict options, the broader the range of possible characters and possible play opportunities that arise. The fewer the rules that restrict options, the more freedom the DM and players have to play the game the way they want.
So what's the point in restricting it? What's the goal?