The fan does not cool the engine, it cools the radiator. If the thermostat is closed (as it is at startup if functioning properly), the engine will get up to operating temp (195) regardless of what the fan is doing, and then the thermostat will open, releasing the hot coolant to the radiator to be cooled. It is at this point when the fan and fan clutch make a difference. The OP's symptoms point directly to a stuck open thermostat. A stuck fan clutch could exacerbate the problem if the thermostat is stuck open, but that is not the underlying cause here.