Yeah the 570 would be better than the 1060. I don't know by how much, but the 470 isn't much slower than the 1060 6GB, even less the 3GB. All about pricing at this point.
UPDATE: I don't follow AMD closely enough to know this but I looked it up... the 570 is about the same as a 1060 3GB but worse in cases where Nvidia cards are preferred. It is slightly better than the 1060 6GB, but again, worse where Nvidia cards are preferred.
So you really need to take it on a case by case basis. They are both in the same ballpark, but one is definitely better than the other one in about 50% of cases, whereas the other is better in the other 50%. It totally depends on what you are playing.