Ignoring price, the Player's Handbook has a lot of sub-optimal weapons; they don't need to be fixed
The Dagger is better than every single 1d4 weapon (Club, Light Hammer, Sickle)
The Light Hammer is better than the Club and the Sickle
Every single 1d6 weapon (Handaxe, Javelin, Quarterstaff, Spear) is better than the Mace
The Spear is better than the Quarterstaff
The Battleaxe, Longsword, Warhammer, and Rapier are better than the Flail, Morningstar, and War Pick
A "bad" weapon doesn't need to be fixed
A lot of these sub-optimal weapons are still cool or useful for character concepts; they just aren't perfectly optimized choices, which aren't things everybody needs to be making at most tables. If you're at a table that requires optimal weapon choices, then a lot of weapons are going to be flat out unusable.
Fixing "bad" weapons takes a lot of work
Fixing these (making each weapon have some reason to use it over the others in the same Simple/Martial grouping) requires significant overhaul and changes to the entire weapons system. I have experience with helping build a system for fifth edition D&D that does exactly this and the system is a lot more complicated than what the PHB uses.
Do what you want and test things out
At the end of the day, a bump in your damage die isn't going to do a whole lot. You're gaining less than 1 additional damage on average (you aren't guaranteed to hit) which isn't very much, it's actually the least amount of extra damage possible at all.
If you're thinking about adding damage or some property to a weapon, I'll always recommend testing it out and seeing how it goes. In this case, I don't believe anything urgently needs fixing (if it does, then lots of other things do too). I would probably recommend increasing the damage die to make it more similar to a Spear, if I had to recommend some sort of change.
Damage type is mostly meaningless
From all the books, the only monsters that are immune to one of bludgeoning, piercing, and slashing and not the others are the Black Pudding, Elder Black Pudding, Intelligent Black Pudding, Reduced-Threat Black Pudding, Ochre Jelly, Huge Ochre Jelly, Sentient Ochre Jelly, and Reduced-Threat Oche Jelly (they are all immune to slashing).
From all the books, the only monsters that resist one of bludgeoning, piercing, and slashing and not the others are the Geist and Topi (resist bludgeoning); the Animated Tile Chimera, Awakened Shrub, Xorn, Big Xorn, Flameskull, Thorny, Vegepygmy, and Vegepygmy Chief (resist piercing); and the Warlock of the Fiend (resists slashing).
From all the books, the only monsters that are vulnerable to one of bludgeoning, piercing, and slashing and not the others are the Skeleton, Skeletal Giant Owl, Skeleton Key, Skeletal Alchemist, Warhorse Skeleton, Blazing Skeleton, Minotaur Skeleton, Ogre Skeleton, Skeletal Polar Bear, Skeletal Swarm, Skeletal Owlbear, Skeletal Two-Headed Owlbear, Thunderbeast Skeleton, Giant Shark Skeleton, Skeletal Juggernaut, Giant Skeleton, Ice Mephit, and Stone Cursed (vulnerable to bludgeoning); and the Mordakhesh, Rakshasa, and Zakya Rakshasa (vulnerable to slashing from a magic weapon wielded by a Good creature).
Unless your campaign is absolutely full of Skeletons or Ochre Jellies and Black Puddings, the difference between bludgeoning, piercing, and slashing damage is almost always meaningless. There are 2020 or so monsters, and only about 40 of them are going to make a difference between damage types even exist.