The problem is less about preventing criticality and more about handling the decay heat from all the isotopes produced when it was critical. That's going to give you 10% of the heat you had been producing with the reactor on for a little while after you stop the chain reaction.
Yes, that has been the common factor in most incidents. Chernobyl being the exception, they didn't manage even manage to get the control rods fully inserted before the shit hit the fan (poor reactor and control rod design).
Still, the British AGCRs were recently fitted with articulating control rods because if the reactor core was damaged they might not be able to insert them normally.
