If you work in heavy engineering workshops black nails are pretty common along with spanner rash of the knuckles.
The doc did my first black nail as a kid with a paper clip after three days of agony.
The first one in a workshop I was basically grabbed and hand stuck in a pedestal drill, 10 seconds later no pain but a bit pissed off. From then on it was DIY with either drill or paper clip.
Haven't had one for years but I wouldn't hesitate to do it if it was still throbbing after a hour.
Good that you went to the doc and the fracture was caught. Yours was different to normal black nail which is usually a squash injury. When the nail falls off you hopefully will have the new one under it. If you don't it takes 2-3 days for the skin to harden and dry out a bit like a popped blister.