One thing I have not seen in all this discussion is, do we know if the ESC and receiver are actually powered up when plugged in to the battery?
1) Disconnect everything from the ESC - the motor and the receiver. Connect a battery to the ESC. Press the power/program button to power on the ESC. Press the button for less than 3 seconds to power the ESC ON, any longer than 3 seconds and the ESC will be in manual program mode. If this happens, just unplug the battery and start over.
2) What do you see? At this point, the ESC LED should at least be quickly flashing indicating there is power to the ESC. If the LED is not flashing, or on in some fashion, then there is a problem with the ESC.
3) If the ESC LED is flashing rapidly, the ESC has power. Next step is to plug the power/signal wire from the ESC into channel 2 of the receiver. Do not turn on the transmitter yet. If the receiver LED is flashing rapidly, then there is output from the ESC to the receiver. The rapid flashing of the receiver LED indicates the receiver is not connected to the transmitter. At this time, that is okay.
4) If both the ESC and receiver have power to them (LEDs flashing/on), then you can test what happens when the transmitter is turned on. Either unplug the battery or use the ESC power button to disconnect from the battery. Turn on the transmitter, power up the ESC and see if you have the flashing LEDs on the ESC and receiver. If the receiver LED is still rapidly flashing, it is not bound to the transmitter and you will need to re-bind.
5) NOTE: I found when I was using a FS-GT3C that it sometimes took multiple attempts to get the receiver to bind to the transmitter. Kind of frustrating and you begin to wonder if there is something wrong with transmitter or the receiver. You do not need the servo plugged in to the receiver at this time.
6) If you make it to this point with power to the ESC and receiver and the receiver is bound to the transmitter, it is time to connect the motor and servo. Connect the motor to the ESC first. Check to see if the motor works when the throttle is pulled. If this works, plug in the steering servo and see if you have steering, or if everything shuts down. That generally indicates the servo is defective.
7) Generally I haven't had issues if a motor fails, it just doesn't work. Bad steering servos can cause multiple issues. That's why when I am having weird issues, the first thing I will do is unplug the servo and see everything else works.
I believe the "beeps" indicating power and system ready are generated through the motor, not the ESC, so you won't hear the "beeps" until the motor is connected. I could be wrong about this but I thought I read that somewhere years ago.
Don't know if this helps? Good luck.