Arduino Mega 2560 Only Powers on with USB
Looking at this Arduino Mega 2560 schematic, if IC1 or IC2 failed (assuming only one or the other is installed for a given board), then you would only be able to power the board using the USB port.

Looking at this ramps 1.4 shield schematic it is assumed that the RAMPS 1.4 shield is providing 12 volts through D1 to the Arduino Mega 2560.  And that testing included using a known good Arduino Mega 2560 board with the RAMPS 1.4 shield board with good results.  Thereby eliminating the possibility that D1 on the RAMPS 1.4 shield is damaged.

If these assumptions are correct, this reinforces the likelihood that either IC1 or IC2 has failed on the Arduino Mega 2560 board.