Most birds will migrate whether or not people have bird feeders out. I too may notice a few birds here and there in the winter time, but still not nearly as many as in the spring/summer. I read somewhere once that sometimes birds will have late clutches of eggs and some of the younger birds may not be able to migrate, which may have been the case with the bird that you found. Your bird was probably able to fly around to look for food, but not old enough/strong enough to make a long migratory trip, or could have possibly been sick/injured to begin with..

And as far as the adapting of some of these birds, some of them (the one that comes to mind offhand is the Mallard) are designed to be very weather hardy. Though they also are designed to migrate in search of better food sources, they also have the capabilities to live in some harsher weather should food sources remain available. So the birds, ducks, geese, etc. that stay, do stay because they have reliable food sources from people, but also have the capabilities of being able to live in colder climates. So I would not worry too much about the ducks, geese, etc. that stay and don't migrate. They stay because they are able to.