So, after multiple tries in dirty things, I decided to reinstall windows.
The last things I tried were :
- Copying the entier c:\Windows from a fresh install to my not-booting install. Windows booted, proving that the problem really came from the OS partition. Unfortunately I could only see a black screen with my mouse on it. I couldn't do anything.
- Copying only subfolders or subfiles of the c:\Windows from a fresh install to my not-booting install, like the c:\Windows\boot folder or things like that. But I couldn't find what to copy to make it boot.
If I had time and patience I'm sure I could have found the files that were broken but I needed my PC and I had nothing to lose on my hard drive so I just reinstalled windows.
Actually I could do that right at the begining but it was way more fun and instructive to try to fix it instead.
Thank you again for your help.