Good deal on problem resolution. I would consider myself very lucky that additonal/significant damages did not occur and that nobody was injured as a result of this problem. Another thing to check for: if your service panel is a Federal Pacific Electric (FPE), Stab-Lok style, I highly recommend it be replaced as soon as possible. Like Aluminum wiring, these panels and their circuit breakers are proven safety and fire hazards. Also, I would have your electrician go through your service panel and check all connections, e.g. wiring to breakers, breakers to bus structure and bus structure connections, check for signs of insulation failure, overheating, melting, look for dark or blackened terminals, insulation, bus, etc. And if the electrician can test the breakers for proper operation, that would be advisable. You may want to consider replacing your breakers if they can not be readily tested or at least replace the breakers that serviced the failed electrical circuits.
A few questions or comments were raised in this thread regarding "why did the circuit breaker fail to operate or open?". Below are some failure modes I have seen during my time.
Mechanical operating mechanism failure, e.g. failed closed, preventing the mechanism from moving or operating to the trip or open position. These mechanism operating failures were a big FPE problem.
Worn or high internal contact resistance caused by corrosion, erosion, contamination), marginal connection between movable and stationary contacts or worn contacts or thermal stress and mechanical loading. Any of these conditions will create a high resistance condition with increased heating effects as current load increases or is sustained. The failing interconnections overheat at high current loading, and, in the worst case, fire ignites within the panel due to insulating material failure.
Defective or failed internal trip unit/sensor (bi-metallic element); fails to sense/trip on overload condition within UL specified time limit to 135%/200% of rated nameplate value. This is analogous to the condition of "overfusing".
Protection system or device trips incorrectly (or fails) due to a fault outside of the protection zone. This means the protective device was improperly specified or applied to provide for proper over current protection and selective coordination. The over current device must be able to operate during fault conditions and isolate the fault to the nearest protective device within its time/current protection zone (long, short and instantaneous).
Best of luck with correcting this mess and if you need any help, shout out.