BTW, if you'd like to see some of these issues "fixed", check my Manual of Arms: Weapons (currently on sale!).
However, we could go even further than fixing weapons, by making them more fun, and a bit different from each other.
In 5e, this is usually achieved through feats. Feats are optional, however, and it wouldn't hurt to boost martials a little. Some of the most interesting feats (Polearm Master, Shield Master, Spear Master, etc.) use bonus actions (or reactions, but we will leave that for another post) to do that. Bonus actions are also what "light" weapons are about:
What if we gave EVERY weapon something to do with bonus actions? With one caveat: Two-Weapon Fighting is quite weak after level 5. So, maybe these options are ONLY available after you get Extra Attack (or "only beyond level 5", like cantrips). Maybe we kill two birds with one stone: make TWF stronger after level 5.
Even better, we don't add any complexity in the first few levels, giving warrior types more options as they level up, but do not make the game more complex for other classes. Notice many of these options will make a warrior with extra attack have two DIFFERENT attacks!
Here are some ideas. Each weapon would have their own trait, some could be up to choice. These are weaker than most feats.
Light (modification): when you use your bonus action to "attack with a different light melee weapon that you’re holding in the other hand", and you have Extra attack, you can do it once per attack until the end of your turn. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
Bash (any non-light, bludgeoning weapon, maybe axes): you can use a bonus action to bump your damage die by one step in your next attack (d6 to d8, d8 to d10, etc.). A maul could cause 2d8 damage!
Armor piercer (could be every weapon except swords etc.): +2 in your next attack if the opponent is wearing armor.
Extend (spears): add the "reach" property with a bonus action (maybe pikes etc. will extend 5 feet further).
Parry (shields, maybe quarterstaff): add +1 to your AC until the beginning of your next turn with a bonus action.
Duck (crossbows and some non-heavy ranged weapons): add +1 to your AC until the beginning of your next turn with a bonus action.
Speed (any light weapon or non-heavy sword): You can make one Attack with it as a Bonus Action on each of your turns. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
Sweep (any big, slashing weapon): When you take the Attack action, you can use a bonus action to attack a different target within reach.
Sword-and-dagger (rapier, longsword): When you take the Attack action and attack with a 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.
Anyway, this is just a rough draft for now, but I think it is a promising idea.