Speedfan is capable of completly stopping a PWM controlled fan, it is not likely but possible. Speedfan is not at all engaged when back in the bios though, many Motherboards have similar speed control software. In the Asus (I don't know gigabyte) software it is possible to have a cpu fan or a chassis fan (non PWM) completly stop, when cpu is cool enough.
It is possible for the 2 controls to be in conflict, With 2 softwares controlling the fans (speedfan and gigabyte softwares).
The "chassis" fans or other fan connectors are not usually PWM driven, although they can have 4 pins (weird), so there is a difference there.
So in your hardware monitoring section of the bios, disable fan control (full speed) , and Disable speedfan from starting and re-test.
To disable speedfan from effecting things temporarily use Autoruns and disable it there and boot. Speedfan is not so easy to setup exactally right, it takes me hours to get it perfect (advanced fan control) and even after that a lot of live tweaking. If your having troubles, you should disable it right away , and test without it. Speedfan can work with other fan control software operating , but in this situation it would be a terrible thing to have on effecting your results.
Software for the hardware (gigabytes own controls) could also be disabled in both the bios, and its drivers in the OS (assuming you could find them) in autoruns (which can disable almost anything).
Right now with no fan going at all, you need to get back to nothing controlling it and it staying full speed always, then to go from there.
Depending on a lot of different factors it is possible even without high temperatures that the lack of properly operating fan could cause the system to fail. Intel has some really nice thermal safety and throttling, but even the application of "turbo" is boosting the speeds, and with that engaged, the sytem could indeed reset. Thermal safeties are there for emergency :-) not normal operations.
It is also possible to have a poor thermal connection to the heatsync, that allows the system to operate, but keeps burping the cpu to throttling or failing. Many people have fixed a odd cooling or throttling or failing situation with the CPU by just properly and more carefully remounting and checking the Heat sync the thermal compound, and its mounting sytem.
I will slightly disagree with your assesment that cold cpus do not need cooling, on the basis of they go from stand-by (C-states) to roaring at full speed in microseconds. So without some cooling or some place for the heat to go temporarily it should just shut down, in microseconds :-0
If it is before you get to the OS software operating of any sort, then you go direct to the bios. If it is after the OS loads , then "anything goes" at that point :-)
I would not be so quick to believe that the hardware itself is failing, when there are so many possible factors involved. One of the ways people will reduce software factors would be to boot into another system temporarily. "it works in linux" :-) Then it is some software in windows that needs adjusting.
other stuff: a 4pin fan is not nessisarily a PWM controlled fan, a PWM controlled fan can be nessisary on some boards CPU fan, usually it is known because it will complain that it didnt see it working. A non PWM controlled 4pin or 3pin fan can usually be made to work at full speed, depending on the design and the hookup of it, I have not bothered with that, just making sure i have a real 4pin real PWM fan for my CPU (at least).