The gun is only one small part, if YOU don't have proper lead and hit the bird it doesn't matter what gun you have. For your question,
I have used an 870 3.5" shotgun for years, also a Mossberg 835 pump action. This year I am using a Browning A500. The new Browning A5 is about $1,500 but he two pump actions I mentioned are $600.00 or below.