The datasheet first page mentions the module has Class B EMI compliance without external components.
The block diagram in the datasheet says there is an EMC filter on input.
If you already pass the required EMI/EMC tests for the class of tests you need to pass, you don't need an extra filter.
Those capacitors are normally used for EMI filtering. If the design is not sensitive to external EMI and not emitting EMI that needs to be kept away from mains (to pass EMI regulations), you do not need them.
Its only that if you decide you need capacitors for input filtering, they need to meet those specific safety requirements, to avoid shock in case of ...
This didn’t fix the issue. Is a varistor with a crack in the case a
sign that it’s also toast?
It's not a good sign and it may well be broken but, given that both varistors mentioned are across the supply i.e. they are rated to withstand 120 volts AC power voltages, then it's unlikely that replacing it will fix the problem i.e. something else has blown.
A crack in the surface or coating of just about any components is an indication of stress, mechanical, thermal, electrical, etc.
So it's probably a safe assumption that your cracked varistor is bad. What is not clear is which OTHER components were damaged by the same surge that took out your varistor.