My spare green CH341A almost always was unable to find a chip. Luckily I have another working CH341A, so started comparing them with a multimeter :
When a "healthy" CH341A is simply plugged into a USB port, without any chips connected, there is a 3.15V - 3.3V voltage between GND hole and other 7 holes at 25xx socket. But a "glitchy" CH341A, when put to the same situation, did not have a ~3V voltage difference between GND and CS pin.
Then I checked a resistance between CS leg of CH341A chip ( found it using this document - https://www.onetransistor.eu/2017/08/ch341a-mini-programmer-schematic.html ) and CS pin at the bottom of programmer (instead of CS hole for convenience) - it was almost always "infinite", but sometimes - when I pushed CS leg with a multimeter pin - there was a close to 0 Ohm resistance, almost zero.
So I did a small soldering of CH341A CS leg, see a picture https://i.redd.it/mfrgs45beuv21.jpg - luckily this CS leg is at the corner of chip so was quite convenient to solder - and now this CH341A is happy and working perfectly! ;-) Will flash many BIOSes :3