I cook things in a pan on the hob, sometimes for an hour or two, and end up with a thick black layer on the bottom. If I used a cast iron pot, would that still happen? 
The situation you describe is burnt material all over the base of the pan. That is caused by using too high heat and would not be helped by using a heavier pan.  The point of a heavier pan is to spread the heat more evenly across the base. A very thin pan would effectively transfer the heat of the burner directly to the food. It would be very hot where the burner was, and noticeably cooler away from it, and food would be likely to burn at the hot spots.  With a thicker base, the burner heats parts of the base but the heat spreads throughout it before reaching the food. That means there are no hot spots, so the food won't burn unless the whole pan is hot enough to cause burning, which it sounds like yours was.  One way to think about it is that a thick or thin pan doesn't significantly affect the average temperature but the whole of the thick pan will be at the average temperature, whereas the thin pan will have hot areas and cool areas.  In your case, it sounds like you need a lower heat, more frequent stirring and possibly more liquid, if that would be appropriate to your dish. (Liquid can convect and disperse the heat through the food.) 