First, always use a soldering iron with a grounded tip. Wipe it on a wet sponge, apply some new solder to the tip, and shake any excess off the tip. Now you'll get good heat transfer.
The trick is add more fresh solder to the joint (thus adding more flux and melting the entire joint on both sides) and then use the vacuum plunger tool to suck everything out. All the solder will flow together and magically disappear into the vacuum tool.
All the boards from that era were hand stuffed, and sometimes they used a tool that both smashed the lead (thus preventing the components from falling out during handling before wave soldering) and cut off the excess lead at the same time. if the lead is flat and moon shaped right where it passes through the board, you'll have to dyke off the tip, or cut the lead on the component side.
After you take out the component, touch the via with the soldering iron and surface tension will pul any whiskers back down to the via. insert the new component, solder in place, allow to cool. Use acetone to remove any excess flux. Throw that copper solder wick away, it isn't worth a damn.
