Assuming that a servo plugged into the slot works OK, that eliminates everything except the ESC.
If the motor turns on switch-on means that it gives a twitch but has no response to the stick, that is no more than a power-on twitch. Not at all sure that that should happen. Does it give the same behaviour if the ESC is left unplugged before power is applied?
It could be that the control part is not responding. Several possible causes, apart from a dead chip.
a signal wire not connected through, chip doesnt see signal
b +ve wire not connected through, chip not powered, can't do anything with the signal
c -ve wire not connected through, signal heavily interfered with.
On ESCs with an external BEC, the main battery ground line can have more than one path to the ESC, and this has been known to cause problems if one becomes disconnected. Does it use a BEC or an RX battery?
A servo tester also does a good job of providing a control input to an ESC.