Why is Arduino Pro Mini not powering this LED?
This was my mistake. I had attached the Arduino to the breadboard by pushing in the provided pin strips, and then putting the mini on top of those, assuming that since the pin was through the hole and looked to be touching, a connection was made. It wasn't.

Rather than continue to mess around with fiddly connections, I just took the plunge and soldered the entire thing to a circuit board. The LED and the button work perfectly as expected, and now I just need to attach the servo.