I shoot them regularly too. I use a shotgun when checking bird houses and when possible walking in the pasture but that is all long range as the birds get pretty spooky.
More often I use the 17 HMR and snipe them from a distance as they wander around the cattle. They seem to be less able to realize I am a danger at 100 yards. They do flee when one of them gets hit but I can shoot them all summer long if I am careful. One has to wait until the cows are clear but that elapsed time may be what causes the disconnect between me showing up and the demise of one of the birds.
They are also quite edible- I use them as "filler" in stews. They would probably work well in dove recipes too. Starlings are the "star" of the old nursery rhyme, "5 and 20 blackbirds baked in a pie...".
