Update:
I also tried a different, known to work, USB-C cable with the same results.
The battery was totally discharged: 0 V between the poles according to my multi-meter.
I tried to plug a lab DC power supply at 3.8 V to the PinePhone battery connector (without the battery plugged in): the phone tries to boot (0.25 A drawn, led turns on green for 1 s.) then it immediately stops (when the LCD tries to turn on it seems), the led switches off and the power drawn falls to 0 A. So the phone is not dead !
Perhaps a capacitor would make it work, I had not one available .
Then I tried to charge the battery by plugging it at 4.0 V (internet says 4.2 V is the maximum tension to charge a 3.8V lithium battery).
After 1 hour the battery was a bit charged, and the phone booted normally
Still not charging though, but when the USB is plugged to the computer, it is seen as a "PinePhone" so my theory that the USB connector is not correctly connected does not hold.
Any ideas ? If you want me to measure tensions inside the phone, I would be happy to do it (if I still keep my 1 month warranty).
I also tried a different, known to work, USB-C cable with the same results.
The battery was totally discharged: 0 V between the poles according to my multi-meter.
I tried to plug a lab DC power supply at 3.8 V to the PinePhone battery connector (without the battery plugged in): the phone tries to boot (0.25 A drawn, led turns on green for 1 s.) then it immediately stops (when the LCD tries to turn on it seems), the led switches off and the power drawn falls to 0 A. So the phone is not dead !

Then I tried to charge the battery by plugging it at 4.0 V (internet says 4.2 V is the maximum tension to charge a 3.8V lithium battery).
After 1 hour the battery was a bit charged, and the phone booted normally

Any ideas ? If you want me to measure tensions inside the phone, I would be happy to do it (if I still keep my 1 month warranty).