"The car has a single coolant temp sensor so the temp gauge would not
work if the coolant tem sensor was bad."

This is not true. If you have never changed the CTS, change it to see
if your problem is solved. The original design CTS was made of resin,
which cracked due to the high heat of the coolant. The new, replacement
OE part is made of brass for durability. If the sensor is cracked, it
will not read the correct coolant temp (in your case, lower than it
really is) and the PCM will not command the fan on.
A bad CTS doesn't always set a DTC, either.