When running any atv with knobbies on pavement, if you go fast enough the shimmy will turn into a flip over.
Not true at all. As I've said here before, where I live it is legal to ride the paved county roads and at times you will see more 4 wheelers on the road than cars and trucks as it is a way of life here. If you have a handling problem it is there whether you are on dirt or pavement. It will only be more pronounced on pavement. Either way it needs to be fixed.
The Bruin is probably like most. It was set up with excessive toe in, in some cases as much as 2 inches, yet no one has taken the time to set it up properly. Give your family car an extra 1/2 to 1 inch of toe and see if you can keep it on the road at speed.