With that kind of read I call villain every time, I _might_ fold if he has only been that aggressive with (close) nuts like a set.

Did you really think that he would've played his KK this way. Normally check-calls on a board like that are signs of monsters. Quite often if a solid opponent calls there after my bet I tend to check the turn in order to keep the pot relatively small even with TPTK and prevent him check-raising me on turn.