I know everybody has their own style but for a board with corroded leaky caps I have had way more luck with gently pushing down on the old caps and using a screwdriver to twist a little bit left then right and back and forth probably 50 times and then the cap comes off at the legs.
I tried two soldering irons, soldering tweezers, even ChipQuik but like you said the cap goo makes it very difficult to heat up the old solder. At the same time, the cap goo makes the capacitor legs very brittle and prone to breaking off. Since I started doing this I haven't lifted a single pad (prob 15+ machines) and it takes about 1/20 of the time.