The tree will not die because of the leaf miners. Yes, they must exert a terrific herbivore load on the tree, but as we can see from trees on the European mainland, which have been mined for 30 years, they still come into leaf, blossom and produce conkers every year. What may happen, over time, is that trees become weakened, and so mortality to other diseases and stresses may be increased. Also, mature trees are much more able to withstand insectivore attack than seedlings or saplings.