I always cut the legs/pins with a fine pair of wire cutters.
Then I heat up each leg on the solder side of the board, pushing gently downwards and then use a solder brush on the other side of the board to flick the piece of leg away. Or, I use a hemostat, clipped onto the leg, hanging under the board. Melt the solder and the hemostat and piece of leg drop out.
At this point I have removed the legs entirely and I'm left with blocked holes.
I'm too slow with the solder sucker so gave up on that years ago. I use desoldering braid instead. Works brilliantly for this sort of thing. I clean the board with isopropyl alcohol and dip the braid in flux, get some fresh solder on the iron tip and get desoldering.
Now I will tell you what my greatest desoldering secret is...
Extra heat!
My Antex iron is 50 watts but have found that where the board has big copper traces, these can suck the heat away like a big heat sink. I use a halogen goose neck desk spot lamp with a 50 watt bulb in it. I let the board overhang the workbench where I'm going to desolder and adjust the lamp so it is under the board and perhaps half inch away from it. This has been very helpful for keeping the board warm enough to prevent heat sinking and allows the solder to flow and wick better. Try it and see the difference.
I have yet to get and try a hot air station for removing sockets but would like to try it sometime.