The problem with your assumption is that formal checking is only that: formal checking. Someone could write gibberish and translate it into valid logic.
Say (in the style of Alice in Wonderland):
If a snark has carrots, then its father is a boojum.
I found your snark with carrots in its ears.
Hence your snark is a son of a boojum.
Real life is not a mathematical world (where you can have such a thing as "something is demonstrated", "something is impossible", etc.) but a physical world where we have educated guesses and at best quasi-certainties (and it is impossible to prove that something does not exist, which is a good reason why we have presumption of innocence, otherwise no one could ever be acquitted of an accusation).
In doing "scientific reasoning", we need to keep into account the applicability of axioms, as well as the soundness/demonstrability of the theorems. Futhermore, there are many subjects where there is no agreed-upon axioms or unambiguous laws (which is why we have debates, conciliations, arbitrations and courts).
You could have such a "logical checking" only when two parts in a debate are agreeing on axioms, definitions, theorems, logic rules and the procedure for verifying facts. But then, if that was the case, what would be the point of having a debate? We would just have to put our question into a computer program and get an unquestionable (or at least unquestioned) answer.
In real life, deduction is only one of two forms of reasoning, and it is usefully applicable to (alas) only a fraction of the phenomena we find around us. In the the immense, marshy, bushy or virgin territory that surrounds our human knowledge, we have only induction as our compass. And that is largely out of limits for computers, let alone logical notations. Which is why we have human researchers in mathematics, physics and other subjects (as well as forensic detectives), who are paid to make educated guesses that others can put to the test.
Sadly, education programs in mathematics are so engrossed with the (actual) wonders of deductive logic that they often forget to mention that, in the imperfect state of our knowledge, it is not always applicable to real life.