I suspect the program wouldn't last long once they realized the beavers would "come back". Not the same ones, but new ones to the same place they removed them from.
Beavers move too wherever they can find a 'good' spot to build a dam and lodge. If your back yard is such a spot, beavers will move in. As the article says, the right solution is to find ways to live with them.
In my childhood home, the big pond behind our house (maybe 200ft across?) existed because of a big beaver dam. Generations of beavers maintained it. We'd lose a tree or two occasionally, but well worth the benefits.