Its more that thermohaline currents drive the gulf stream - Hyposaline warmer water rises, moved towards the poles. Hyper saline cold water sinks, moved towards the equator creating a 'conveyor belt'. As the polar ice caps melt, the water becomes less saline, stopping or slowing the 'conveyor belt' and therefore the gulf stream. And as you've said, us being on the same latitude as parts of Canada and Russia = snow and cold in winter way beyond what we're used to.