You never use the soldering iron set at a temperature close to the melting point of solder. The copper pads on the circuit board and the component itself would suck up the temperature from the tip and the tip itself may have a very small "reservoir" depending on the shape of the tip... so you pretty much always have to use slightly higher temperature that the solder melting temperature.
As for electrolyte boiling at 100c and all that talk, the metal can of the capacitor, the leads, the pads and traces on the circuit board will basically act as a shield/heatsink, preventing the contents from reaching so high temperatures so fast. You have a few seconds to put solder on a capacitor pads and leads without the electrolyte inside being damaged.
63/37 solder isn't "low temperature" at least the way i see it. Low temperature solders imho would be those Chipquick solders, those that have bismuth in them and they melt at around 138 degrees Celsius and stay semi-liquid/liquid for a longer period of time.
Anyway, another suggestion for desoldering such capacitors would be to get a hot air gun (could be something cheap as a paint stripping gun) and heat up the pcb under the capacitor to around 60-80 degrees then add a drop of flux on the ends of each capacitor and use proper tip (for example one of those cup style which can hold some solder in a depression, used for drag soldering) to put some solder on the pads. Just by heating up the solder by moving the tip from one side to another, you should be able to quickly push the capacitor off its location.
This technique is often used to repair broken LED backlights on monitors, where they use aluminum circuit boards - they heat up the boards from the bottom to a reasonable temperature in order to minimizer the iron tip temperature effect on the plastic case of the leds.