Electronic – Quartz crystal gasket, isolator, platform or whatever

Quartz is a piezoelectric material, which means that if you mechanically deform it, it develops charges on its surface. Similarly, if you place charges on its surface, it causes mechanical stress in the crystal. The way a quartz crystal benefits a circuit is that mechanically the crystal acts much like a tuning fork, with a natural resonant frequency, and the piezoelectric property allows that to be coupled into an electronic circuit. Since the resonant frequency is mainly determined by the physical size and shape of the quartz, you get a frequency reference that is much less sensitive to temperature than you would get using just LC circuits.

Your problem is almost certainly not a direct issue with the solder itself but with the flux in the solder. Many fluxes leave a conductive film on the board which must be removed for circuits such as yours.

Note that RTC chips try to be extremely low power so as to preserve the battery as much as possible. As such, the biasing of the 32.768KHz crystal for the oscillator is designed to be rather low current (i.e. high impedance circuit). The flux conductivity can sometimes be low enough to completely swamp out and literally short out the oscillator preventing it from working.