Monks get 10 darts as part of their starting equipment. Except it looks like they are only proficient in Simple Melee Weapons and Shortswords. Did I miss a rule somewhere?
There are two different relevant entries in the Monk here. The first is their proficiencies. The second is what counts as a "Monk Weapon." The dart qualifies as the former, but not the latter.
First the Proficienies list:
Weapons: Simple weapons, shortswords (PHB 77).
So yes! The monk can use the dart. However it doesn't count as a "Monk Weapon"
...monk weapons, which are shortswords and any simple melee weapons that don't have the two-handed or heavy property. (PHB 78 emphasis mine)
You can see the difference in these two lists. Monks can use the dart as it is a simple ranged weapon. However, they cannot use them with the Monk's martial arts class feature since they are not melee weapons.