Finder seems to be broken, Desktop files missing
Massive thanks to @Tetsujin. It worked. 

Just a small tip if you experience the same issue, the re-install seemed to be stuck at '1 minute remaining'. I dug a little bit and the system was copying everything from a temp folder to the final destination. 

You can activate the logs by pressing Cmd/L or just wait until it finishes (it can take several hours/days).