Unsalted butter will most definitely burn. All cooking fats will burn, given a certain temperature.
If you want to use butter, but want to cook at a higher temp, use clarified butter (ghee). The smoke point is 485F, which is quite high. The solids in regular butter will burn at a much lower temp.
I use to use way more oil whenever I was doing anything, its easy to go overboard. I like frying tortillas as a quick snack in my 10 inch cast iron fry pan at home and only about 1 teaspoon is plenty enough to fry both sides and make it nice and crispy. I usually go for vegetable oil when I do that.
My bad, you are right. The butter should be clarified first.