B737NG Fire Detector Fault
 

Going to my QRH recently and noticed the following in the fire section:
For the APU Fire Detector INOP, it says: shut down APU and CAUTION, do not operate APU anymore because a fire would not be detected!!!
For the Cargo Fire Detector Fault and, more importantly, for the Engine Fire/Overheat Detector Fault it says: the detection system in one or both of the cargo holds/engines is inop....
Period, end of checklist!!
This seems a bit odd to me. Is a running engine with an undetected fire less dangerous then a running APU with the same problem? Consistency would suggest to shut down the engine and land asap, or at least land asap on both engines.
What if you get an engine fire detector fault indication in the cruise with 2 hours of flight to go to your (engineering equipped) destination and ten minutes away is another engineering base.
Continue to destination or divert??

Greetings, Ziggy

APU - single loop
Cargo and engines - double loop ????

You cannot use the APU because you will not know if it is on fire, not a good situation.

If you lose both loops of the fire detection system on an engine, you still have 149 fire detectors in the back of the aircraft, half sitting on the left, half sitting on the right.

As far as shutting the engine down, after learning that it is on fire I would be inclined to execute the engine fire, severe damage or separation checklist recall items. That should tick that box.