Iron heats up briefly then quits

The power light is in parallel with the heating coils. The heating coils are in series with a thermostatic switch (the temp control) and a high temp cutout. So, it is either the switch, cutout, or coil in that order. All that requires disassembly to locate the cause and fix it. However, like hair dryers, the only thing that might be replaceable is the switch. The other parts are welded in place because of the heat generated.