So... here's what I'd do if I were narrowing weapon proficiencies to the 5 categories with really simple understandings.
Split Bludgeoning, Slashing, and Piercing into 3 groups for each weapon type that have their own levels within the Basic through Master proficiency. And let people rank up their proficiency with a type (Bludgeoning, Slashing, or Piercing) as they gain levels and/or feats to do so.
Basic: B1d6, S1d4, P1d4
Simple: B1d8, S1d6, P1d4
Martial: B1d8, S1d8, P1d6
Advanced: B1d8, S1d10, P1d8
Master: B1d10, S2d6, P1d12
Bludgeoning: Any weapon meant to smack someone with no edge or point. All Bludgeoning weapons gain the Versatile weapon trait.
Slashing: Weapons meant to cut someone with a bladed edge. All Slashing weapons gain either the Versatile or Finesse weapon trait.
Piercing: Weapons meant to stab someone with a sharp point. All Piercing weapons gain the Finesse weapon trait.
So for someone like a Wizard with only Basic Weapon Proficiency, the d6 Bludgeoning Weapon (Quarterstaff, Club) would work fine. While a Rogue with Simple proficiency might prefer to use a Shortsword (1d6), a Dagger (1d4) or an Iron Shod Club (1d8) for their Sneak Attacking.
Of course, a Fighter with Master Proficiency and that Shortsword will hit for 2d6.
A Rogue who loves Daggers could work their way up to a 1d12 dagger by taking 3 extra levels of Weapon Proficiency: Piercing.
Is it particularly good at simulation? YMMV. But it'd be super simple, you must admit.
It also means two very important things: A weapon's type is much less important than the skill of the person wielding it (A Dagger in the hands of a Master is deadly, and a Katana in the hands of a Novice is far less so), and weapons themselves are less mechanically diverse while being significantly more narratively diverse. In that you can describe any Versatile Slashing Weapon as a Longsword, Shortsword, Battleaxe, Sickle, Etc.