Well, this isn’t what she said. It is true that given the choice, many horses will choose to walk around on a cold day rather than stand in a stall. But obviously that depends on so many other factors - wind especially, and precipitation. And in a place like Western NY - snow depth can make walking much more challenging.
In an ideal scenario, horses would have a mix of options. But, an insulated and/or well constructed barn that blocks the wind properly should be fine. I took lessons at a place that had a metal barn - on cold days - it was definitely not warmer IN the barn than outside. Especially if the sun was out and starting to warm things up.
If I had to consider a move to a cold, snowy climate with an elderly horse, I would definitely consider retirement boarding in a warmer climate. I live on the other side of NY and get far less snow, but when we’ve had large snowfalls – it is hard. Hard on the horses, hard on the barn owners/help. A light, fluffy snow is fun. Two feet of heavy, wet snow is not. A good setup makes a huge difference, because shoveling paddocks/gates can be very tough.
I definitely would be very careful about committing to buying property, building a barn, and managing an elderly horse at home versus boarding in a setup that is prepared for the work for the climate.