MacBookAir7,1 Early 2015: MJVM2LL/A (1.6 GHz Core i5 or 2.2 GHz Core i7).
Went completely dead after coming back for a short time once more. It looks that something happened already after I've taken the battery out - the unit went live on charger alone, although with some hesitation. I didn't know what G3H was and googled it, found that there are two: PPBUS_G3H and PP3V42_G3H, didn't have time to explore more, but it looks that the problem is not simple and may be not fixable by me, although I've done a lot of small electronic repairs with some success.
After the charger was out for some time it takes 30 sec or much more. The green light comes back fast if I just take the connector out and immediately put it back - about 3 sec. The answer is helpful for better understanding.