I don't have the ability to open the board design files but it looks to me like the 32KHZ crystal http://www.mouser.com/ds/2/137/1697497-472682.pdf
I am sure that is used for the RTC. While there would be some difficulty, at least it is a largish two-terminal device so you might be able to repair it.
I don't know much about crystals but maybe pressing on it was enough to simulate ticks to cause the system to start.