The correct response is both: brake hard to shift weight to the front, and then immediately steer to avoid the obstacle. By shifting weight to the front you dramatically increase the grip on the front wheels, helping you swerve quickly. It also has thr benefit of slowing the car if there is an impact.
Presumably the car would be able to do this better than a human.
My concerns about a self-driving car would be handling winter conditions: poor visibility, snow collecting on sensors, variable traction conditions, no lane markers visible, etc. I'd be astonished if a computer could figure out how to get a car unstuck even if only slightly encumbered.