"Why do people in the US and Canada think winter starts now?"

Because now is (very roughly) when it starts to get cold, on a regular and consistent basis. Just because it's the shortest day of the year doesn't make it the coldest day of the year. There is a connection between sunlight and the temperature outside, obviously, but it's not as direct as you might think. The main cause of this is water.

Oceans, lakes, and even groundwater collect an unbelievably immense amount of heat during the summer. That keeps the temperature somewhat regulated during the summer. This is why sometimes August is the hottest month even though the longest day of the year comes almost a month and a half earlier, in June(!) During that whole time the water is still absorbing all the heat and keeping the temperatures slightly lower than they otherwise would be. The reverse happens towards the shortest day of the year, as all the water is slowly letting off all that heat and keeping the temperature up. It may snow or there may be chill winds (because the upper atmosphere has long since cooled off) But overall it doesn't really get cold until the water has released all of its stored heat, and the limited daily amount of sunlight is all that's left to provide heat.