PTC's used in degaussing circuits are capable of handling, in some cases, 30-35 amps initially, and some also contain an auxiliary heater. The standard rating time is 3 seconds for the cutoff, and the current flow at that point is only 200-300 ma. The idea is to pass the inrush current but then the resistance increased dropping the current. At only a few hundred ma, the coil only has a small field. Curve steepness is relative, and yes, I know there are many types of this thermistor, and consider a 3 second cutoff rather short. Not exactly fast switching as electronics go, but faster than a linear graduated thermistor as you be used to monitor temperature.
My only point was that I prefer to energize the coil for more than the typical 3 seconds, and withdraw the subject tool slowly from the field before the current is altered. Just a personal preference.