I have the same problem. My 2013 MBA battery failed in 2019, which I diagnosed by disconnecting the battery and discovering it restarted on mains power. Replacement battery from Replacebase lasted 2 years before swelling up, but during that time, whenever MBA was fully closed down or ran out of charge, I had to unscrew the back and disconnect battery before it would restart. Now in 2021 onto my third battery (£65 “genuine” Apple/SMP battery from eBay … haha), that problem just got worse, as (a) it won’t re-start when it’s cold (<15 celsius), and (b) discharges more easily as I avoid over-charging for fear of battery swelling/fire.
I had to replace the hard drive in 2020 with a 3rd party (Sabrent) drive, which definitely messed up the sleep/hibernate modes, as system usually reboots instead of sleeping. However, since the latest battery change, that might also be why it’s fully closed down. (It’s hard to be sure of exact cause as my 11-year-old would never admit if she let the battery run out etc).
Any ideas why it won’t restart until battery has been disconnected? A couple of people over at Stackexchange also puzzled by this issue. I doubt it’s the power supply board. Please don’t say SMC reset etc. I suspect sleep/hibernate software conflicting with weak battery power delivery.
(I just managed to restart it from cold after it initially failed to respond to power button: I quickly closed it, warmed it slightly by hand and it booted the moment I attached power lead… interesting!)