So some feedback here.
After starting and testing this helicoil fix, the leak was still there. I looked into it again and found a hairline crack in the water pump cover, along the threaded hole. Long story short, the cover is shot.
I remplaced it by the NOS one, new (metal) gasket, new screw, new crush washer and all is fine now.

To celebrate the end of this annoying episode, I went for a nice ride! 😎