It’s not counter-productive at all! The more, the better.
It drives me nuts when people short-change the cool-down, so good job for making it a priority. I think there’s an aerobic benefit from cool-downs because muscles are forced to keep working when lactate is still high. So they have to work while reabsorbing lactate. I think there’s something in Olbrecht’s book (The Science of Winning) about that.