Keep in mind that I hail from a farming background. Woodchucks a.k.a. groundhogs a.k.a. "Mr. Buttons" are one of the most destructive rodents anywhere. They are responsible for property damage, crop decimation, and hazards to livestock and equipment. They rank right up there with House Sparrows, Lamprey Eels and Asian Carp. If you don't mind them destroying your property then to each his own. In Michigan, they can be taken year 'round, unlimited for the cost of a small game license. Same holds true for House Sparrows but no license necessary.
OK, back to the OP's topic. Are the native birds smaller than a House Sparrow? I made a sweet repeating trap for sparrows that works on a counterbalance principle. Smaller, lighter birds are unaffected but a HoSp will trip it and he has but one way to go...into the holding area. My trap can hold dozens of sparrows and I even provide them with food and water while they await their fate. Kind of like the band playing while the Titanic sank. I built mine from a picture I saw on the net and it uses a Backyard Chicken growers staple....1/2" hardware cloth and scrap lumber.
I'm sorry if I offended anyone, but when I grew up, there were clear boundaries and definitions of profitable and unprofitable, destructive and harmless, and they were dealt with accordingly.