Radiator fans will not turn on automatically to keep the engine cool?

Well, from the description it sounds like the low-speed relay isn't operating properly. The way the system works is that once the coolant gets up to the lower temperature it turns on the low-speed fans, in that mode the fans are run in series which means they both get approx. Half battery voltage. Then once it heats up to the 212-degrees high point relay three switches and grounds both fans and now they run at high speed. The relay block is on the left fender liner. To test them and the wiring you can use a simple jumper wire at the dark green and dark blue wires that control the relays. If you ground the dark green wire, it should turn on both fans on low speed. If you ground the blue wire, it should kick the fans up to high.
Some scan tools would allow you to test both systems while others only turn them on high speed.

For some reason part of my reply is missing.
You need to test the control wires first. If grounding those wires turns on both speeds then the problem is the PCM, if grounding the wire for the low-speed side (dark green) doesn't turn them on then you can test in the relay socket itself to see if it's the relay or the wiring to it. So first take a couple pieces of wire and connect them to a good ground. Then go to the PCM and at connector 1 find the dark green wire at pin 42 as well as the Blue wire in connector 2 pin 33 You can use a simple sewing needle to pierce the insulation on the wires to be able to connect for testing, when you are done some liquid tape or similar can be used to seal the wire. Now use the grounded wires to test the system. With the key on touch the Dark green, do the fans come on at low speed? Now while holding that connection, ground the blue wire. Do the fans now switch to high speed? If both wires do what they are supposed to do, you need to replace the powertrain control module. However, if you have a good scan tool that can activate both speeds you can do the same test using it instead of poking the wires.
Now if your earlier testing turned both fans on at high-speed relays 1 and 2 were working. If it didn't turn the low speed on, then you replace relay 3 and test low speed again.
Be sure to check the three fuses for the fans as well.
As the fans came on in your earlier answer it isn't the fans, it is in the control circuit to them. Either the relays or the PCM. However, without knowing if the scan tool tested only high speed or only low speed you need to test a bit more to find the real cause.

No rush.