How should I take out (or fix) this broken potentiometer?
You are basically correct, but not quite. It's true that you need to desolder the pot pins then replace the pot. However, I'll pretty much guarantee that all 3 pins are connected to the circuit, so you can't just "leave the pin alone". 

And in this case, desoldering is made easier by the state of the pot. Take a pair of sharp cutter, and cut pins 1 and 3 at the body. Then melt the solder on pin 2 and pull the lead and body away from the board. Then unsolder 1 and 3 while pulling with a pair of needle-nose pliers to remove them. Now take either some desoldering wick or a Solda-Pullit and clean out the holes, and install the new pot.