Actually, it turns out there was some leak damage, around the battery connector.

I tried cleaning with isopropyl alcohol, but it still shows the problem.

I tried checking the traces and many are good, I couldn't check them all because I'm not sure the isopropyl is cleaning all the rust and some pads seem covered with it.

I'm also concerned about some white stuff around the pinss of one big chip on the bottom. Cleaning removed some of the white stuff, but not all, some seem under the chip.

I was wondering if there are more in depth cleaning methods to remove the leak residues, maybe some vinegar ?

By the way, would anybody have close up photos of the same region of a working motherboard ? That could help me see if some traces are totally gone

Ouch, that’s rough. That sort of damage is really hard to repair if it gets into the vias. I don’t have advice to offer, just wishing you luck.