1. I'm responding to
"Nothing you just said disputes any of what Peterson has said. ... try[ing] to discredit him without actually addressing or disputing anything he's said shows you just have an axe to grind with no substance."
Nothing there about his status as a public intellectual.
2. "Anybody can pick a few random quotes out of context and dispute them." These are tweets he sent on Twitter, not replies to other people. There is no more context. There is a high density of this crap on his twitter feed. At the time I looked, the quoted tweets were in position 4, 8, and 17 on his feed. I could have disputed more of them but those were the easiest to address.
3. There is a difference in between "women definitely tend to be more attracted" and "Women deeply want". If Peterson meant the former he should take his own advice and be more precise in his choice of words. Similarly if I've misinterpreted his other tweets, that's on him for not providing more context. He's not new to this game. (And this is, apparently, one of his rhetorical tactics. There is link in another comment to "bait and switch".)