Seems like the system partition wasn't broken but user partition had a problem (the kindle wouldn't index new books before, partition was often mounted read only by my friend's OS because of errors...). He tried to reboot hoping it would index some books. The kindle got stuck on the tree of death, with lighting on. Sticking on a wall charger for a couple of day made it reboot into diag mode.
It was then possible to access the user storage (wiped clean by the kindle when it was stuck on the tree: that was not a boot loop, it was actually partitioning the buggy user partition) and an upgrade (usb mode / latest .bin from amazon /
exit diag mode) was enough to get the device to reboot in a clean / stable mode.
Now I'm not sure what would be the best way to check if the nand is in good condition or not...