MOVs typically fail by the "breakover" voltage decreasing as the small internal "crystals" (particles) short due to higher energy "hits". If it gets low enough, it starts to conduct during regular line current peaks, and soon fails altogether.
UL requires MOV surge protectors to have thermal and overcurrent protection*. Mostly the whole house ones have lights on each leg, AFTER the protection. So if it fails, the protection opens, and the light goes out.
The protection is sensible. I have seen MOVs that simply disappeared. Part of the case was still there, but the "works" had vanished, vaporized by the heavy current when they failed. When I say "vanished" I mean that the "works " were still present, but in the form of a dirty (and probably conductive) coating over the rest of the stuff in the unit. That's undesirable......
An MOV can handle a LOT of peak power, but not for long. It's intended to cut the peaks off short transients, Such as ones typically used in testing (and representative of reality). for instance 1.2 x 50 microsecond or 8 x 20 microsecond pulses. They are NOT intended to protect against longer overvoltages from causes like connecting to wrong voltage, faults in the power grid, etc.
* The MOV can draw a small current at full voltage, which is the usual endpoint of their life. Even though it is relatively small, it can overheat the device, which may only be the general size of a US "quarter", say 23mm or so diameter.