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?
I see Simple Weapons under monk proficiencies, not Simple Melee Weapons. You should be fine.
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.
Simple Melee Weapons and shortswords are monk weapons, not proficiencies. So you can still get your proficiency bonus with darts but you can't use your monk abilities with them.