Thanks, I'd hope I was correct, it's my job to know :-P
As mentioned before, the battery is only required to make the RTC store the date/time information when the power is off. If the system is turned on, the RTC is powered from the power supply, and the battery charges up at the same time.
Even if the PCB tracks appear to look clean, it's still possible one could be open circuit due to corrosion. This usually happens in or right beside vias (the holes joining tracks between PCB layers).
It's also common for corrosion to damage U177, the RTC data latch.
My website, mentioned before, has all the information you need to fix the RTC, look in the RTC repair section and follow it through step by step.