This is wrong on so many levels that it's difficult to keep track of them all. Let's start with the introduction:
Why did people who deployed these [critical thinking] terms always look so rigid, so predictable, so feeble? Why did people who avoided them look so confident in comparison, so much more in command of their resources, so much more mature? Their arguments seemed to possess an inner strength; the baloney detectors, by contrast, only had strength in numbers.
Translation: Some people never shut up after they have been proven wrong, and the author values this assertiveness over whether their arguments are logically sound. He concludes that the most assertive arguer is automatically correct. He calls for the total abandonment of logic and reason and to respect force and effect in their stead.
His justification is the presumption that any person would quit an argument after being proved wrong; this is a false premise, as evidenced by the existence of liars, fanatics, and human choice. He applies this false premise to the whole of reason in order to condemn it; this is a strawman argument, a form of false premise, as reason has never presumed that its methods would lead to this result, but holds that its methods enable a witness to more accurately judge whose arguments are more likely to be correct. From there he reaches the conclusion that critical analysis is worthless. In an irony, he is himself attempting to use critical analysis to debunk critical analysis itself, but his foundation is so rotten that his argument is easily shown to be unjustified.
Next the author tries to declare certain forms of arguments as being off-limits to critical analysis, like emotional appeals. An emotional appeal is irrelevant to whether the stated facts are likely to be true or false. The dismissal of an argument for its emotional appeal is shorthand for saying that the argument has not justified its claims, but is trying to trick witnesses into believing its claims by appealing to their emotions instead of reason. If an argument truly has nothing behind it other than its appeal to emotion, then its dismissal as an appeal to the emotions is fully justified. He also applies the same protection to the Rush Limbaugh defense of people saying their lies, initially presented as serious factual claims, were meant to be taken as "comedy" when disproven, or as "irony, hyperbole, understatement, whimsy, counterfactual conjecture, or any other of the wonders of figurative language that defy semantic nit-picking". Contrary to having "nothing to contribute", "semantic nit-picking" tears these turns of phrase apart and exposes the argument being made.
Next he says that the correct identification of a logical error does not imply incorrectness of the argument. No comment is needed.
Finally, we get to the root cause. The author has constructed this framework of deliberate mental incompetence to justify supporting Rebecca Watson's claim that all Atheist men are sexist, with the implication being that Atheist men are sexist to such a degree that all of them are rapists-in-waiting. A supporting argument is laid out in a similar manner to the classic form, simplified as such:
Thesis: All Atheist men are sexist.
1. Z is an atheist man.
2. Z is not sexist.
(Points 1 and 2 are repeated for a dozen more Zs)
It takes the complete abandonment of logic to conclude, as the author does, that the thesis is upheld and indeed strengthened by these arguments. As we have seen, the complete abandonment of logic is the author's goal. He wants you to believe that you should stop thinking logically because he writes so strongly and forcefully for it.