MacBook air not turning on

Replace motherboard, that's the option bcos it's probably a processor failure from the motherboard

This does sound like a logic board fault but it is actually very difficult to confirm other than by elimination. It can be normal for the fan to momentarily spin when the power is connected to the machine and before the machine it turned on. It could simply be that the signal from the power button is not reaching the logic board. Every logic board has power-on solder pads that can be used to manually power not he machine by shorting them. You would have to Google where these are on your specific model, but this can rule out a faulty power button.

Disconnecting anything non-essential helps eliminate other components like the Airport/bluetooth board & battery.

Beyond that, the only way to be sure is either to replace the logic board or get your logic board fitted to an identical model MacBook Air that works but this is obviously not practical for most people.