Just to reassure yourself, here's what I would do.
1. Check the operation of the aux fan by unplugging the temp switch on the right side of the radiator. With the ign. switch on, but not running, and A/C off, jumper the brown ground wire to one of the remaining wires. remove the jumper and then jumper from brown to the other wire. The fan should run in both cases; one jumpered pair should make it run at a faster speed than the other. If the fan fails to run for either test, something is wrong with it.
2. Check the operation of the temp switch itself. Remove the top radiator support bracket and slide a large piece of corrugated cardboard down in front of the radiator so that it blocks airflow across the radiator. You should be able to leave the support bracket off for the test. If the radiator will not stay in place, wire or zip tie it in place as,when the time comes, you will want to be able to remove the cardboard without delay. Now start the car, A/C off. While monitoring the temp gauge, run the car until the temp starts getting to the mid-zone and then start checking the aux fan to see if the switch activates it before reaching 3/4. Once it does, go ahead & pull the cardboard so the rad can cool back to 1/2 before shutting down. This will also confirm that the temp switch is properly turning off the aux fan as well. (If you wanted, you could take the temp to max to confirm the fan goes to high, but I don't think I would do that on my car. I'll just assume that if my gauge ever went past 3/4, I would be stopping the engine before reaching those extreme conditions and not risk breaking something during the test). Conversly, If you don't see any fan action by the time you hit 3/4 to 7/8, pull the cardboard and let the engine fan cool down the radiator, and go buy a new temp switch.