Chosen Solution
I was gifted a broken MacBook Air early 2014. I was thrilled when it fired right up, only to find that it would randomly shut down. Once shut down, it won’t power on right away, but will if left to sit for a bit. I’ve tried blasting it with a fan tray to see if it was a thermal issue, and also removed the logic board and cleaned with alcohol. I’ve also done SMC and PR resets. The behavior seems to be unchanged. The maddening part is that it will work briefly (or sometimes longer), but eventually always shuts down. It never starts up again right away, but eventually (after a long wait) will power on again. The power on is also slow - if I sit and watch with the bottom cover off, I see the fan spin up, stop, spin up, stop and finally spin up , chime and boot.
Your Mac is powering on correctly when it does start. This model starts and stops the fan twice before booting. You would seem to have a problem with a corroded trace or oxidised solder balls or pads under one of the BGA chips that is why it is shutting down and only starting sometimes. This is a frustrating problem but put under a microscope could well show signs of where the problem is. Of course I would rule out a parts issue first like battery, DC board, I/O cable and track and touchpad.
