Maiming only the cambium layer in different spots shouldn't put the tree to death because the saps would still flowing in the sapwood and in the bark, giving him time to overcome that and heal. Actually, the tree kills himself by isolating the thousands of small wounds. He closes the vessels, fill the cells with shit and abandon the area in the hope that will stop the intruders. As usual for any wound he can suffer from. But here, there's just too many. The opportunist fungus invading the wound and adjacent tissues don't help either.
In France, the ash is heavily affected too. Not by the EAB but by a fungus killing the twigs, then the limbs, and eventually the whole tree (by contaminating the stump from the fallen debris). It's the "chalarose", coming from up north. Many many trees have a hairy look due to the replacement sprouts successively killed in the crown. Big amount of dead wood too.