I do have some thoughts, but I have to preface what I say by emphasizing that I have NO extensive knowledge or experience with the refrigerator--just what I picked up from researching and repairing my own cooling problem. I didn't even know exactly how the refrigeration cycle worked on these combination (gas and electric) fridges. I do have a little experience with conventional refrigeration systems--but I found that these are quite different, and the system is more complicated.
The first question would be how does your unit work on electric power? Your post detailed LP operation today, but didn't mention electric. The key is that the primary difference between gas and electric operation is only how the ammonia/water mixture is heated in the boiler (electric heating element or gas flame). In my case, my unit would work fine on electric power, but not cool when switched to propane. That told me that the basic refrigeration cycle was working normally; that whatever issue I had was specific to the gas flame--and as I said in the other message topic I eventually found a blockage in the gas heater exhaust vent that was not allowing it to heat effectively even though the actual flame looked generally OK to my untrained eye. For instance, the condenser cooling that we've been discussing (air flow, fans, thermostat switch, etc) has to happen for the fridge to cool on either gas or electric--so if either one of the two works, condenser cooling is not likely to be the problem.
However, if your refrigerator fails to cool essentially the same on both electric AND propane, the problem would likely be in the basic refrigeration system--which is common to both. That calls into question several parts of the system--the thermistor you mentioned, leveling problems which lead to blockage of the system from the liquid not properly draining, inadequate heating of the boiler for some reason, inadequate condenser cooling from poor air flow (baffle, fans, thermostat switch etc), or even control system problems (board) or coolant leakage (not likely).
To directly address a couple of questions that you asked: First--yes, the factory fan could be wired to an on/off switch and bypass the thermostat---BUT consider this: there are many, many of these fridges out there working at least adequately, with the original setup. My own fridge worked perfectly again on propane (in some very hot conditions) once I corrected my problem. If you are certain (or are able) to establish conclusively that your problem is tied to condenser cooling, then modifying your factory fan to run full time would be one part of the solution (I'm not aware that there can be a problem from over-cooling the condenser but I don't know that for sure)--but I think you should try to be sure you're working on the actual problem before you spend more time on it. Second--the fact that the factory fan shut off after you started your external fans sounds like that part of your system is working properly. As I mentioned in the earlier post, my information is that the thermostat switch is supposed to open back up at 105 degrees F to kill the fan. Sounds like your external fans simply dropped the temp below that level after you started them and the switch opened as prescribed.
Sounds like you have some extended experience with this refrigerator, so I'm sure you already know that an absorption fridge does not react quickly to cool-down. I don't know what the internal temps were when you fired it up, but the 53/65 that it reached would be doing well for my unit if it was starting at 90 degrees--maybe not even as low as you report in four hours. At least you have a remote thermometer to tell you what is actually going on inside--that's one of the handiest tools I've found in my own effort to keep up with mine. If there is any other info I might provide, please let me know.
All the best, Mike