Let me tell you where I'm coming from: Yes, I've seen all the textbooks with their unending treatments of logic ... Ugh! Sure, logic is important, but why indeed the insistence on formal logic proofs? We have logic courses for that. Indeed, just the fact that the focus is on deductive logic seems rather silly. Most real life reasoning is not deductive and so all this logic has very limited applicability. Yes, logic teaches one to be careful and organized in one's reasoning, but let me put it this way: when it comes to people coming to bad beliefs and making bad arguments, logic is actuallyprobably the least of our problems. Much more problematic are our cognitive and social biases. When you cover fallacies, you'll find that they can almost all be traced back to those biases, rather than to any logical reasoning impairment.
Statistical/inductive reasoning: really important stuff! And by the way: I take a very 'anti-math' approach here as well: you can do all the probability theory and statistics you want, but in the end so many people are still convinced by some hasty generalization of refutation based on one piece of personal evidence andexperience or anecdotal evidence: that's the kind of crazy thinking that I want my students learn to prevent.
Causal reasoning: super important! The way we think the world causally works translates in how we act. And my oh my, are there a great number of pitfalls ofin causal reasoning!
Oh, and in case you think I'm all anti-math and don't like logic: I teach Introduction to Logic, Intermediate Logic (meta-logic; soundness, completeness, etc) and Computability and Logic (undecidability of logic, Godel results), and I love it! ... So yes, I could certainly teach a whole critical thinking course all around logic ... it would certainly be a lot easier for me to teach (and assess!) than what I do now ... I just don't think itlogic has all that much to offer as far as critical thinking goes.
Sorry, all more than 15 years old I'm afraid, but at least they go beyond mere arguments: there is a lot of psychology and sociology behind the formation of our beliefs, and you have to instill awareness of that in the students, or else in my eyes it fails to be a critical thinking course. If there is one key to critical thinking, it's independent thinking!