crystal oscillator of 32768 KHz
This is a typical frequency for a clock.
The DS1670 has microprocessor monitor circuitry (pins 13 and 20). The handling of touching some components is probably interfering with the monitoring on that chip and generating a reset. I suggest you monitor the signals around it to see if these behave as described/expected in the datasheet. Cost of one is 11$ at digikey, that is an expensive chip...
Pin 13:
Active-Low Reset. The RST pin functions as a microprocessor reset signal.
This pin is driven low 1) when VCC is outside of nominal limits;
2) when the watchdog timer has timed out;
3) during the power up reset period;
4) in response to a pushbutton reset.
The RST pin also functions as a pushbutton reset input. When the RST pin is driven low, the signal is debounced and timed such that a RST signal of at least 250ms is generated. This pin has an open-drain output with an internal 47kOhm pullup resistor.
Pin 20:
Active-Low Strobe Input. The strobe input pin is used with the watchdog timer. If the ST pin is not driven low within the watchdog time period, the RST pin is driven low.
Another thing in this chip is that it is capable of switching power sources (to maintain clock and settings memory I think). There might be an issue around the battery. Not charging or bad contact or corrosion maybe, and when you touch the board it all of a sudden works just long enough for the thing to reset and start.
POWER-UP/POWER-DOWN CONSIDERATIONS
The DS1670 was designed to operate with a power supply of 3.3V. When 3.3V are applied within nominal limits, the device becomes fully accessible after tRPU (250ms typical). Before tRPU elapses, all
inputs are disabled. When VCC drops below 2.88V (typical), the RST pin is driven low. When VCC drops below the lower of 2.7V (typical) or the battery voltage, the device is switched over to the backup power
supply. During power-up, when VCC returns to an in-tolerance condition, the RST pin is kept in the active state for 250ms (typical) to allow the power supply and microprocessor to stabilize.
It indeed sounds like some capacitor issue or a bad solder joint. Something that is sensitive during start-up but not anymore after that.