Welcome to the forums. Yes, the enrichment valve is supposed to pull out with the cable. Sometimes, they will seize in the carb like yours did. When it seizes, the cable will just pull loose like you described. You may be able to spray it with penetrating oil and let it sit and it might come out later, but chances are that it won't. If you pull too hard, the top where it hooks to the cable, will just break off. The best way that I've found to remove them is to drill a small hole down through the center and screw a screw down into it and grip the screw with a pair of Visegrips and pull it out. You will have to have a new enrichment valve.
If you don't want to buy a new valve, just leave it stuck down in the carb and you can use the primer button on the side of the carb to start it when it's cold. Just pump it 3 or 4 times and it should start.
Most 4-wheeler problems are caused by a loose nut connecting the handlebars and the seat!!
You only need two tools in life -- WD-40 and duct tape. If it doesn't move and should, use the WD-40. If it shouldn't move and does, use the duct tape.