For an ugly DIY hack you can sometimes gently rock the electrolytic capacitors back and forward (in the same axis as it's two pins)
The goal is to pull the capacitor off the pcb while leaving its legs behind on the motherboard.
You then have two metal legs sticking up about 10mm which you can solder a new capacitor onto.
This way you don't have to unsolder anything.
Another option, if there is room, is to solder the new capacitors onto the back of the motherboard in parallel with the dead one. Electrolytic capacitors fail to an open circuit state so there isn't a problem leaving the dead one there.
Just make sure you get the polarity correct and use low esr caps.
Quickly soldering capacitors on the back of the PCB is something i do temporarily to confirm a bad cap fault.
If the product starts working again i then solder the cap in properly.
It just saves time in case the board has other problems, otherwise you can spend hours replacing all the cap for a totally dead board.