Teaching logic is vital to any form of critical reasoning or argument construction. It would have been an excellent idea except that it doesn't fit with the current 'teach you what we want you to believe' curriculum. Be it the current 'science is everything' position, or pastafarianism, the approach fails when it attempts to dictate what to think.
Logic itself is insufficient as it can only confirm the logical validity of an opinion, never its accuracy. I'll illustrate by following your reasoning: you hypothesise that we can improve reasoning by examining bias; you assert that humans can come to conclusions without bias; you assume the rules of mathematics apply (reasonable I agree); you apply those rules to a mathematical problem to propose that a statement is 'simply true'; you extrapolate that we therefore can devise statements that are incontrovertible (that is, 'not false'); thus you prove your assertion; then extrapolate to prove your hypothesis for all statements made without bias.
Logic demonstrates the validity of your reasoning, showing each progression. It does not demonstrate that the final position itself is correct, only that it is not incorrect. Logic may demonstrate that I am not incorrect in believing evolution, pastafarianism, or love, but not that the statement 'all should worship his noodleage' is correct.
I think that if people knew how to think, you would have more of the PETA etc groups as everyone would have their own, logically valid positions on every issue. The issue isn't the logical process (although that weeds out some pseudo-scientific positions), but that logic requires a final position that is either true or false, and we have chosen to reject the idea that any source may dictate or define absolute truth. This requires us to 'prove' the absolute truth on any issue, which is only possible by testing every hypothesis - which is impossible.
Improving logic skills would be excellent, but until there is proven absolute truth (currently only possible in maths), all logic can show is that you are validly uncertain.