We see firsthand the Elementals protecting the whole forest in the blasphemy quests. Without their help,
the forest wouldn't have healed from the blasphemy there and we and the Elder Seedseer would have died from the fake creeping death.
As far as the Ala Mhigans, they invaded the part of the Twelveswood that now lies in The Fringes zone, on the east side of Baelsar's Wall but west of the river. That part of the forest doesn't look as healthy so it could be the elementals were never that strong there. The guardian tree there and the elemental inside it are dead as well, but that seemed to have been the Garleans' doing when it was poisoned by Black Rose.
As far as getting rid of them, you can't. It was established back in 1.0 that elementals are in EVERYTHING and are EVERYWHERE, not just the Shroud. That's just where they're concentrated the most. There was even a quest where you went to the Hearers to have them ask the elementals in the ink of a letter what they thought about the person writing it. Gridania probably is the least populated of the Eorzean city-states. I can't imagine it's particularly easy to feed a population when you can't have vast fields of wheat and need to be careful with anything botanical. So the whole thing affects relatively few people and those who don't like it leave or go angry athiest and get petrified.
The Gridanians are the ones who want to live there and living with the elementals is part of it. The elementals were there first, people came later. If people living there have a problem they can move to literally anywhere else. No one is forcing them to be there and the whole country's existence is tied to them. It's like complaining about Dravania being "infested" with dragons. People who go into someone else's home, put their feet on the couch and say it's their home now are called Garleans.