Chosen Solution

This is a i7 1.7GHz, 8gb RAM and 250gb SSD. Suddenly it would not start up. I suspected the I/O Board since the charging light was only sometimes on. So I replaced the I/O Board and it started up. The problem now is: Will not start when battery is connected even if charger is attachedStarts fine if I disconnect the battery and start from chargerConnect the battery and the battery charges fineDisconnect charger and runs fine off of batteryRestart works fine from battery onlyWakes from sleep from battery onlyShut down and the issue is back, will not start and have to disconnect battery to start from charger What I have tried: Tested another working I/O BoardReset PRAM and SMCRun Apple Diagnostic - passed all testsTested with another working batteryTried to jump start from start pads behind the WIFI board Is there some kind of chip that triggers from the battery that is bad? Any suggestions? Here is the trackpad cable

Update (07/19/2018) No, @danj I really don’t know what is bad here. I have tested with a working I/O Board cable but with no better luck. I can start with shortening the pads but only when the charger is connected. Just randomly I can start with the keyboard power key but only if charger is connected. I looked at the I/O cable connector on the logic board and it looks different from another (2013) MBAir in the way that on one side there are 2 pins less. Not easy to explain but I could not get a good picture of it. Could this be any issue? I measured the the start pads. 3.41 V without the charger connected and 3.42 V when the charger is connected. Not big difference but a differnce. I don’t understand the logic, I am thinking this is some sort of grounding problem. Without proper ground there will not be a shorting? But the grounding is obviously not in the trackpad and its cable. When the charger is attached there seems to be grounding since it starts then. Is this a path to investigate?

The trackpad is the terminus of the keyboard cable and then both the keyboard & trackpad signals travel over the trackpad cable. What I often find is corrosion on the trackpad and/or the cable. Pull the trackpad out and do a good inspection looking for corrosion. MacBook Air 13" Early 2014 Trackpad Replacement You likely need to replace the trackpad and one or both cables.