I generally like the weapon approach in 5e so I'd only make minor tweaks:
1. Make all crossbows simple, and all bows martial. Even the hand crossbow; I don't see what the big deal is.
2. Add a "Min Str" column, and give many weapons a minimum Strength score to wield them (or else you get disadvantage). For example, the heavy crossbow might have Min Str 15 -- giving the light crossbow a reason to exist. (Ditto for longbow/shortbow.) If you don't meet the requirement you have disadvantage on the attack roll. This would fix several other quirks (like you could slap a Min Str 11 on the rapier and nerf a lot of Str 8 builds), and may allow you to make some other weapons simple.
3. I also like the idea of Versatile being a composite property that can affect things like reach, finesse, and even Min Str. For example the longsword having Versatile (d10, finesse) and the spear/quarterstaff having Versatile (d8, finesse) are great ideas.
4. Make the flail Versatile (d10), to make it cool for huge dudes in armor wielding a ball-and-chain. And I'd introduce a "light flail" that is martial d6, finesse, light -- basically a bludgeoning version of the short sword. Monks would become proficient with light flail and scimitar, so that they're not all using short swords, spears, and quarterstaffs. While we're at it, the war pick needs to be Versatile (d10) also, because those things were actually pretty cool.
5. Add the scythe, a slashing greatclub.
6. Make the trident Versatile (d10, finesse) which puts it into a really weird niche, as appropriate for a really weird weapon. But, I'd clarify that thrown weapons can only be thrown with one hand (you can't two-handed throw a spear or trident).
7. Make the whip a light weapon, because that's awesome.
8. I like the idea of an "off-hand" property for daggers that lets them be used in TWF even if the weapon in your other hand isn't light.
9. Bump up the light hammer up to d6 for parity with the handaxe.
10. Add "shield" as an entry, in its own category (separate from simple, martial, etc.) to clarify what happens when you bash with it.
Finally, I'd change the Weapon Master feat considerably, because as written it's a crap feat:
- You become proficient in all simple and martial weapons.
- When you hit with a weapon (not an unarmed strike or spell attack), you deal +1 damage.
- While you are wielding a weapon, you have advantage on any ability check or saving throw you make to avoid dropping the weapon, being disarmed, or otherwise losing your grip on the weapon.
- When you wield a simple weapon, increase its damage die by one step: d4 to d6, d6 to d8, d8 to d10, and d10 to d12.
(Basically, it makes any weapon you wield "martial," plus a +1 damage bonus on top of that.)