MacBook Air won't power on, except I remove the battery and plug it back in (SMC reset?) - sleep works
There are two possible causes these symptoms:


Faulty Battery
Bad SMC Chip (Logic Board)


Faulty Battery.  As the OP described, once the battery was disconnected, the machine booted fine.  What could be happening here is that the battery life has degraded to the point where it will not hold a charge above the threshold to power on the device.  

If you can power the MBA in this manner, try resetting the SMC then reconnect the battery.  If it continues to fail, you probably need a new battery.  If you already know how to open the MBA and disconnect the battery, you are halfway to replacing it.  If you're not sure, ifixit.com has an excellent tutorial.

Given that the OP's machine is a 2012, it would mean the battery is pushing 5 years now which is around the EOL for this battery.  It's time for a replacement.  I recommend getting one with at least a 18 month warranty.



Faulty SMC.  There is a possibility that the SMC (chip) is faulty.  The first thing to attempt to diagnose it would be to run Apple Hardware Test (AHT).  Hold the D key while booting from a powered off state with the AC adapter connected.  

It may come back with "no errors" so the positive diagnostic test would be to attempt with a known "good" battery.  If it continues to fail after the battery is replaced, there's a high chance than the SMC is bad.  It's time to take it in for service.