I'm confused how the oscillator speed would have any effect. Like Psi pointed out, it has poor pixel response time. Probably a function of the on-chip FETs used to switch the pixels?
If you slow down the video you can almost see the individual pixels turning on at different times.
Another thing to check is the voltage/waveform at which the crystal is being driven, you might need to improvise something thin enough to slip under the zebra strips to the COM/SEG ITO. Your module might just have the crystal voltage tied to Vcc but the HD44780 allows a separate voltage for LCD drive, which can go up to 11V.
What is the ambient temperature? LCD response time is negatively affected by low temperatures.
Edit: KS0066 allows up to 13V for LCD drive voltage, but it must be
below Vcc so the bias voltage should be connected to a negative supply.