I'm not an expert on the issue, but is difficult to believe they would destroy the ecosystem, unless it is an ecosystem already made dysfunctional by human intervention. Herbivore population is limited by food supply. In my country they would probably come in hand to clean the woods and prevent fires.
Usually, when wolves are removed, and deer overpopulate; the area becomes a wasteland. I watched a documentary where they said that a lack of predators caused destructive behavior changes in herbivore populations. When predators are present; deer, elk, etc., move out of areas quickly. When predators are not present; they eat everything until there is nothing left.
Overgrazing by deer is changing the face of U.S. forests
earthsky.org
Scientists in the U.S. Northeast published two studies examining the impact of deer overpopulation on natural ecosystems in early March 2014. Biologists at Cornell University investigated disruptions by large numbers of deer to natural growth in developing forests. University of Pittsburgh researchers showed how large deer populations are causing an increase in garlic mustard, an exotic invasive plant, in forest understory flora. In both instances, the root problem is overgrazing by deer of native plants that open up more growing space for invasive exotic plants that deer find unpalatable.
These studies were conducted in in Ithaca, New York and Pittsburgh, Pennsylvania. But the problem of deer proliferation is widespread across the U.S. and Canada. Their numbers have increased dramatically for several reasons. Since the arrival of European settlers more than 300 years ago, the deers’ natural predators, wolves, have been exterminated. And, as human populations have increased, deer forest habitat has shrunken drastically, mostly giving way to suburban lawns, gardens and farms that can also provide a deer’s food sources. Compared to historical population estimates prior to European settlement, deer populations today have increased, depending on location, by four to 10 times.
Under normal conditions, when cleared land is left to revert to forest, there’s a natural progression of vegetation: grasses are gradually replaced by herbaceous plants and shrubs, followed by native tree species such as cottonwood, locust and sumac. Deer, however, have a strong preference for consuming native plants over exotic invasive plants. As a result, most native plants are unable to recover from the pressure of overgrazing by deer. This allows exotic invasives such as thorny thickets of buckthorn, viburnum and multiflora rose bushes, largely shunned by deer, to become dominant in the landscape. Where deer had free rein, the plots showed patches of bare soil and fewer plant species, especially a lack of woody plants