I don't have any "Sleeve Bearing Brushless Cooling Turbine Blower Fan" in hand, so I will just find any BLDC motor to do the OP's test. Usually BLDC motor has a couple of wires to control speed and direction, so make the test simple, I will just short the PWM/speed and direction signal wires to Vcc, to make it a 2-pin wire similar to the OP's BLDC motor. The BLDC I will be using is the following:

As I found a very good spec on XTW6009 which accepts low Lipo 3.7V/3V input, so I will forget other alternatives like MT3608, LM2596 etc. Now I am going back to do a second/follow up test on TP4056 (with protection DW01 and dual N-channel power MOSFET).

Now I have wired up and testing the following:

1. Lipo 3.7V single cell,

2. XTW6009 step up module, stepping up from 3.7VDC to 12VDC.

3. DC12V DC motor (note: not BLDC, but simple DC which is more newbie proof)