Macbook Air 2013 won't boot up; Charging cable never turns green, always amber

Is there any way of diagnosing a fault with the I/O board? Any other suggestions to look into?

It’s possible that it’s your DC In Board as you stated, however the symptom you describe indicates it may be something worse; it may be your logic board.

• Macbook charger isn't working
• MacBook Pro early 2015 not turning on and not charging

The amber led means that the MacBook has sensed the charger (1 wire charging circuit) and is sending power, but the battery isn’t charging. We know this because it won’t turn on and the battery never completes a charge cycle (light never goes to green)

It’s cheap and easy to replace the DC in board, so that’s pretty much the diagnostic. If that doesn’t solve the issue, it’s your logic board (possibly your SMC, any of the power rails, the battery charging circuitry, etc.)