That fan on the CPU may be faulty. Certainly, for whatever reason it is not starting up and running. Now, on many mobos the CPU cooling fan's operation is monitored closely to prevent possible CPU damage from rapid overheating if the fan fails. In such systems, the mobo may shut the whole system down very quickly if the CPU cooling fan fails to send back to the mobo header a good speed signal.
I asked about "low noise adapters" because IF you use them, they may reduce the voltage supplied to the fan so much that it fails to start up, which is what you see. BUT, you may not be using those things, and the failure to start could be for another reason.
Try this as a test. Do it only briefly - for less than a minute, I suggest. It is to test the fan AND the CPU_FAN header. Shut down. Now, interchange where fans a connected. Plug the actual CPU cooling fan into a mobo SYS_FAN header that already has a fan on it that is known to work. Now plug that case fan that WAS known to work into the mobo CPU_FAN header. Set up this way, the actual CPU cooler will be fed a voltage signal that you know is sufficient to start up the case fan. AND, the case fan you know will work, will test whether it also can work properly when fed power from the CPU_FAN header. Now, start up the system and observe carefully and quickly. Did both fans start up? Did only one start up? If they both started and kept running, you're safe to let it run for a short time because the actual fan on the CPU IS running now, unlike when it was not running before. BUT if that actual CPU cooling fan still fails to start, you must assume it is faulty. On the other hand, if that CPU cooling fan does run when plugged into a SYS_FAN header, but the case fan fails to run when plugged into the CPU_FAN header, you know there is something wrong with the CPU_FAN header itself.
Let that test situation run for no more than a minute, preferably less if one of the fans fails to start, then shut it down. Let us know which fan worked and which did not.
By the way, you did not ask this, but I'll add just FYI. You say your CPU cooling fan is 3-pin, plugged into a 4-pin CPU_FAN header. That still should work. Some headers like that are set to work only in the new 4-pin PWM Mode, and that would make your 3-pin fan run full speed all the time. Others are set to detect which fan type is plugged in, and change to the older Voltage Control Mode if it's 3-pin, but that still should be able to start up your fan properly unless it has been set to something weird.