No.

I can't speak to the entirety of the fur trade, and I've found references to changing fashion (in addition to reduced supplies) killing interest in furs.

But I can say that a lot of the overhunting was done not by Europeans, but by indigenous Americans seeking greater amounts of furs to access European trade goods. The Haudenosaunee or Iroquois, based in upstate New York, engaged in a series of expansive, often genocidal wars that stretched as far as Illinois, motivated in part by a desire to increase access to beaver pelts and in part by a desire to replenish their numbers, hit hard by smallpox and measles, through mourning wars. The volume of this fur trade nearly extirpated the beavers from much of the territory involved, and the Iroquois were far from the only participants - their rivals were themselves trapping fur and selling it to Europeans, as were indigenous peoples nearly everywhere else in North America far enough north to produce decent pelts where European traders ventured.