Yup, same here with my Air 1. The only way it will charge is if I go through this particular routine ........
Lay it flat, plug the charging cable into the iPad *only, lift up on the cable right where it enters the charging port, then plug the other end of the cable into the charger that’s already plugged into the wall, wait until it says its charging *both in the middle of the screen icon *and up in the right hand corner icon, continue to keep the cable end lifted up and slight an object under the cable end, let go and wallah! ‘Simple as that’
Been doing it that way for months. No other method will get it to charge, ya, I’ve tried every suggestion on the internet *except, as you have mentioned, trying to charge it when it is turned off, I’ll give that a go just out of curiosity.
Otherwise it still functions just fine. After a long, long, wait, I finally got to purchase the new Gen 3 version 12.9

It won’t replace it completely but instead will become my daily user while the ol’ Air will relegate to ‘occasional use’.