Just to play devil's advocate: the first algae were extremely damaging to the environment by releasing toxic oxygen. Of course, that took millions of years to take effect.

So beavers could in principle also be damaging, but taking a long time to kill. (And, of course, nowadays a lot of organisms have come to depend on the once-toxic emissions of algae. Beavers might have a similar effect on the ecosystem.)

See https://en.wikipedia.org/wiki/Great_Oxidation_Event