C12 is a power supply bypass, according to the schematic. It's probably not directly the source of the problem. But indirectly, by leaking goo everywhere it may be your culprit! The recap has to be the first step, otherwise you'll be chasing after new and weirder problems as time goes on.
If you've got a multimeter (or a scope) you can probably check the 1Hz RTC output (UK4 pin 1) to see if there's any sign of life.