Growing up in the Midwest, I remember being taught (not sure if this was official or something I got from my father, older brother, or maybe an uncle) to quickly assess if I believed that I would hit the deer or not.
If I was going to hit the deer, regardless of my actions, I was to accelerate into it.
If I could brake and avoid it, I was to brake.
I've been in that situation a few times. In the case of a collision with a deer, I accelerated into it and only took damage to the front passenger side panel. I do believe that it helped prevent a bad situation from becoming worse.
In the other cases, I was able to avoid a collision by braking.
Later in life, I did swerve once to avoid a large animal that I mistook for a person walking on a very dark, very rural road. My car flipped and slid on it's top ~100 ft. before coming to a stop. Myself and all 3 passengers walked away without incident but I still feel fortunate that we all did these many years later. Swerving is almost always the wrong answer.
Reason given was that braking causes the front end to drop causing the deer to be “pushed” up onto the hood and into your windshield.
No sure if this is true or not.