IMHO, the advantage of using a solder sucker or solder wick is that these tools will remove excess solder, and leave a nice, clean eyelet. When you use a toothpick, you are just pushing the solder aside.
I have used a probe to open up eyelets too, and sometimes I do both: remove excess solder with solder wick and then open the hole with a probe.
Whatever you do, be careful not to damage the wires coming from the transformer coil. I had one fellow bring me a transformer that wouldn't work after he replaced the cord himself. He had broken off the wires to the primary coil.
