Hello and welcome to JustAnswer!
If the temperature gauge works ok then you can rule out any problem with the coolant sensor.
The PCM could be a possible cause of the fan not coming on if it had a failed relay driver and couldn't turn the relay on, but that probably isn't your case. It's not common or likely, and in your case if there was a problem with the PCM that would mean you have more than one problem going on at the same time.
If the fan won't come on low speed when you power it from the relay then either the wire from the relay to the fan is broken or the fan motor is bad. Wiring problems are possible, but it's very likely that the low speed circuit in the fan motor is bad. This isn't an unheard of problem on the PT. Since the fan seems to run slow on high speed that's another sign that the motor has failed.
P1684 sets when the battery has been disconnected.
If there are no other codes then the wiring from the PCM to the relays is fine. The PCM itself is also very likely ok.
There are two circuits in the motor, hence two relays, two wires, and a three wire electrical connector on the fan motor. One wire is ground, the other two circuits come from the high and low speed relays and utilize separate circuits and brushes in the motor to give you the two separate speeds.
You're welcome, and good luck with it!