In Netherlands nearly everyone cycles for nearly all local trips all year round regardless of the weather. Nothing beats the convenience of a bike.

I use mine for every trip longer than a 5 minute walk and less than a 45 mins bike ride. That encompasses the whole of Amsterdam and its outlying suburbs.

In bad weather, people have suitable rain gear and accept getting a bit wet. Or cycle with umbrellas. As the Dutch say, "there's no such thing as bad weather, just bad clothing".

Here in the Netherlands during winter, bike paths are cleared/salted before most residential streets. That together with the sheer amount of bicycles keeps all mayor bike paths clear enough, though it does get slippery.

Places with good bike infrastructure take care of it (e.g. plowing).

but it's still cold as fuck

Well sure. I mean, bundling up is possible, just ask anyone in Chicago. My hunch is that places with healthy bike infrastructure are also dense enough to make most trips short, minimizing time spent outside.

Winter in Copenhagen means –5°C, so it's not really the same. It is more humid than the continental USA.

But yes, the cycle paths are ploughed (often before the roadway). Nevertheless, bus and train use increases.

For cycling a long time in the rain, some people use overtrousers or other waterproof clothing.

Just means you can bike faster without sweating.