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.