My toy simplified 5e makes classes have weapon damage.
Simple, Thrown and Ranged weapons do d4 damage.
Wizards, Rogues and similar do d4 with everything.
Rogues get sneak attack, which adds cinditional d6s, and finesse, which adds 1d4 unconditional.
Rangers do d6 with one handed weapons, and can twin strike both twf and ranged/thrown weapons.
Paladins and Barbarians do d6 one handed/d10 two handed.
Fightres do d8 one handed/d12 two handed.
All attacks add level to damage, but level is small.
Barbarians get a growing Rage Die.
Scaling in T2 is done with unique mechanics instead of everyone gets extra attack:
Fighters get cleave and extra attack.
Barbarians get Mighty Blow
Paladins get Holy Sword and Smites.
Rogues get more SA dice.
Rangers get Whirlwind Attack (aoe) and Disruptive Strike.
Calibrating damage is not that hard so very different mechanics deal about the same.
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Giving weapons some kind of property is fun. But it can easily lead to some strange results, like a rapier being the best barbarian weapon.
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A less extreme variant is just replacing extra attack.
Monk: fold flurry into attack action, make it 1 extra attack. Give extra bonus action at 5.
Barbarian: Give an extra weapon damage die at 5, 9, 13 and 17. At 5 grant an unarmed or improvised attack for free. Replace rage damage bonus with rage die, scales from 1d4 to 1d12. Unarmed/improvised can replace weapon damage with rage die.
Fighter: keep extra attack at 5. Grant combat superiority at 5 (enemies in melee range get disadvantage they ignore fighter, fighter gets advantage).
Rogue: Swap uncanny dodge for disruptive strike (reaction to a hit, if it hits force a reroll and halve damage).
Paladin: Holy Sword at 5. Their blessed weapon does +2d8 radiant damage, and gets advantage on foes who have hurt an ally of theirs since end of their last turn. Smites get +1d8 damge at level 7, 13 and 19. Improved Smite is 4+2d8 on holy sword not +1d8. The blessed weapon cannot be removed from their possession while the Paladin is alive.
This makes the Barbarian fight half of the time by picking up stuff and smashing people with it. And every class feels different.
Except Ranger.