Try this technique.
Get the old caps removed. Remove the old solder, so that you have flat pads. Clean the board and remove all the flux residue.
For each capacitor, there is a pair of pads. Melt a little solder onto one pad out of each pair. Just one, not both.
Now, set a capacitor into position. Use a tweezers, if needed. Gently press down on top of the capacitor (use the eraser end of a pencil, or a flat-heaad screw driver or some such) to hold it in place. Apply your soldering pencil to the pad which already has solder on it. Let the capacitor sink into the solder down to the pad. Be sure to heat both the pad and the capacitor terminal with your pencil, so you'll get a good joint. Cold joints happen when one or both terminals being soldered don't reach solder melting temperature.
Remove the soldering pencil. Wait a few seconds. Remove the tool pressing on the cap. Now solder the other terminal, the way you would solder anything else. Apple heat to the pad and terminal. Melt the solder onto the pad and/or terminal. Remove solder and heat.
Rinse. Repeat.
I cannot see how this is more difficult than soldering a wire to the pad. But I may be blinded by my point of view.