MacBook Air does not boot at all, fan spins up momentarily

My MacBook Air (13 inch, Mid-2013) cannot boot all of a sudden. Everything was working fine previously, and no water damage, drop, or other problem arose in the meantime. The machine suffered a minor drop much earlier on that left one corner dented, but no board was damaged.

Whenever I press the power button, the fan spins up momentarily, but clicks to a halt less than a second later as if there is insufficient power to the fan. This behavior is observed in all circumstances: battery power alone, battery + charger, and charger alone disconnected from battery. Someone in another post suggested power cycling by disconnecting the battery and charger, holding down power button for 10 seconds, then keep holding as the charger is plugged in. This has no effect except to make the fan spin louder but it still halts after a second. Under all of the above circumstances, the fan tries again every five seconds until the power source (battery, charger, or both) is disconnected. The same behavior is observed when the logic board power pads are shorted with the keyboard disconnected.

If it is connected to the MagSafe charger, the status LED lights up correctly. When I try to reset the SMC, the LED cycles to green then amber, as expected.

I have disassembled the Mac and the logic board has no obvious signs of corrosion or blown components. The moisture indicators are not tripped and there is not even any significant amount of dust, with none in the fan at all.

I have tried resetting the SMC and PRAM/NVRAM numerous times, but there is no obvious reaction from either: the SMC reset cycles the charger LED as mentioned earlier, and resetting the PRAM/NVRAM has no effect. In neither case does the startup chime sound or the screen (including backlight) turn on.

Needless to say, the Apple Store says it's a broken logic board. I think that a power surge or software update might have been to blame, but I am not sure. I appreciate any insight you may be able to offer.