My proposed fix:
Primitive/improvised weapons:
One-handed melee: 1d4 (club)
Two-handed melee: 1d6 (walking stick, great club)
Thrown: 1 damage (rock)
Simple weapons:
Versatile melee: 1d6/1d8 versatile (mace, spear, quarterstaff)
One-handed ranged: 1d4 (sling, dagger, handaxe, light hammer)
Two-handed ranged: 1d6 (shortbow)
Martial weapons:
Versatile melee: 1d8/1d10 versatile (long sword, battleaxe, morningstar)
Heavy melee: 1d12, two-handed (great sword, great axe, maul)
Reach: one lower die size (polearms: d10, two-handed -- hoplite spear: 1d6/1d8 versatile)
Light: one lower die size, and must be finesse (short sword: d6, light, finesse)
Finesse: one-handed only (rapier: d8)
Two-handed ranged: 1d8 (longbow)
One-handed ranged: 1d6 (javelin)
Javelin: Change from simple to martial weapon
Trident: Remove from weapon list ("spear" can represent tridents)
Staffs: The "walking stick" replaces the old quarterstaff. The new quarterstaff is on par with the mace.