MPG320I is absolutely correct! The auxiliary fan is essential. It not only comes on when the AC is operating, but it's also controlled by a temp sensor in the bottom of the radiator. If the radiator is not cooling the coolant sufficiently, the aux fan will come on and push more air through the radiator, especially at low road speed. I've driven my '81 across the Mojave and AZ during summer and the temp gauge never got above three-quarters of the way to the red with the aux fan running continuously. Some other things to check are the condition of the radiator and the water pump. A previous owner could have used Stop-Leak in the radiator to cure a small leak and clogged up the radiator core. If the water pump has a metallic impeller, it could be corroding away. I had that happen on a Ford F-100 with a 360 V-8. The impeller was corroded to the point where there wasn't enough left of the impeller blades to move the coolant fast enough to maintain proper cooling.
You don't have to overheat the M-10 engine very many times to warp the head and blow the head gasket. So, if I were in your position with a car with an unknown history living in a desert climate, here's what I would do: Remove the radiator, take it to a radiator specialist and have it checked out. Get it re-cored if necessary. Pull the water pump and take a look at the impeller and the bearing seals. If any thing looks questionable, replace the pump. Ditto for the thermostat. If the hoses look old, replace them too. Also, take a look at the plastic shutoff valve for the heater. It's located near the firewall next to and below the brake booster. Replace if evidence of leakage is present.
If the car has the stock AC system, don't get your hopes up about being cool during a Mojave summer. The AC system that was installed in the E21 worked okay as long as it didn't get too hot outside. The one on my '81 was worthless at ambient temperatures above 90F. I finally took it out when it failed for the umpteenth time.
Hope this is helpful.