RE: Motor shuts down moments after throttling up.
I have been flying Electric RC for some time now.
When you Throttle up and the ESC shuts down the motor, it is usually because you are pulling more Current (amps) than the ESC is rated for. I have seen this many times.
Buy or borrow a Wattmeter. This device is connected between the ESC and the Battery. It will tell you how much Voltage, Current, and Wattage you are using. You can put a smaller prop on the aircraft until, at full throttle, you are not drawing more current than the Motor, ESC, or Batteries are rated for.
Check the specs on the Motor, ESC, and Batteries. You need to have the correct Voltage batteries. The Capacity (C) of the batteries determines how long you can run the motor. The batteries will be rated for how much current you can draw Continuously, and in a Burst. If, for example, the battery capacity is 2500mah and the battery is rated at 20C, you can draw a maximum of 5 amps (5000 milliamp hours) Continuously. If you system draws more than that, it will shut down or you may damage the batteries. The ESC will shut down the motor if you draw more current than it is rated for. If the voltage is too high, you can damage the ESC or the Motor. If you are using LiPo batteries, 2cell is 7.4 volts nominal and 8.2 volts fully charged (3.7v per cell), as an example.
Reduce the Voltage or Current Draw or lessen the Load on the system by reducing the Diameter or Pitch of the prop and your Throttle should work normally. If the ESC is defective, that might also cause it to shut off under load.