Technology

The idea of don't swerve for deer is very common and is taught in driving schools. If you've never heard it until today, well you were let down and today you learn. You don't know dismiss it because you haven't heard it.

Swerving is dangerous and even if you think you can do it safely, having a deer appear while travelling at high speeds is risky, even more so at night.

You're supposed to slow down but stay in lane.

The reason you're supposed to swerve for things like Moose is because moose are big as fuck and tall, and if you hit one head on, you will cut the legs out from under it, and it's massive body will roll through the windshield and crush you, killing you or causing massive bodily harm.

This is from the Virginia DMV for example (emphasis mine). Them not having something about moose is actually bad as well.

https://www.dmv.virginia.gov/sites/default/files/forms/dmv39d.pdf

Deer/Large Animal Hazards Tens of thousands of crashes with deer, elk, and bears take place in Virginia each year, resulting in fatalities, injuries and costly vehicle damage. To avoid hitting a deer or other large animal:

• Be alert at dusk and dawn especially in the fall.
• Slow down if you see a large animal near or crossing the road. Large animals frequently travel in groups; there are likely others nearby.
• Use the horn to scare the animal away.
• If a collision with a deer or other animal is unavoidable, do not swerve. Brake firmly, stay in your lane, and come to a controlled stop.

I'd imagine there's a few reasons for the variation in driver training between upstate NY and Wyoming.

1. Road Speed. Here in Wyoming our highways are 65-70MPH (posted) and most of the Interstate is posted at 80MPH. You can generally figure that everyone is doing at least 5MPH over that. The higher the speed the less time you have to react and the harder it is to lightly twitch a vehicle to one side or the other.
2. Road layout. You commented about swerving into the shoulder but most of our highways have a shoulder width of 48" or less and on the other side of the shoulder there's commonly a ditch. It has to do with the wind and snow we get here but if you twitch onto the shoulder here you are likely to encounter a very unwelcome surprise.
3. Animal differences. In upstate New York you're dodging Whitetail deer, here you're trying to dodge Antelope (which are nearly as fast your car) Mule deer, Elk, Black bear, Brown Bear, and the occasional Moose. The bigger the animal the harder it is to dodge.

The way you describe upstate NY is how it was taught to me when I grew up in Nebraska but it's not what they advise in Wyoming. Here you stay in your lane and slow down as much as you can before impact.