I have this problem too with a MacBook Air 13” Mid-2011. Even with the SSD pulled out, when you power it on and press the ALT key to halt any missing OS warning, after about 15 seconds the fans start spinning up really fast.
With the SSD in, the progress bar loads fast and using a Kill-A-Watt meter I can see is draws about 15 Watts, up until about 60% the progress bar slows down and the power draw drops to 7-9 Watts. Take about 5 minutes to get to the login screen. Then takes maybe 15 minutes to show the desktop. I already booted to recovery and ran a check on the SSD file system, checked out okay.
I tried a new battery, and for the heck of it I tried another power I/O board since it refuses to charge the battery. The strange thing is once I was running the diagnostic and decided to pull the power cord, and the battery was still powering the system, so it appears the battery is good enough in some ways, it’s just the laptop immediately refuses to charge and jumps directly to the green light.
I’ve tried letting the diagnostic run a half an hour, but it appears to be frozen. I’ve already tried all the reset options too. My only option at this point seems to be the ewaste pile.