Bad choice of example, but I think I agree with your main point—it’s a common mistake to focus on one component of an argument when it isn’t your main objection, where that fact being true or false will not change your final conclusion much.
However, I’m not a big fan of the general argumentation/”arguments as soldier” context. In such a situation, I would not try to find the argument that best demolishes the other person’s argument, I would try to work out on what exactly we agree and on what exactly we disagree. For example, if someone says “Fact F is true, therefore we should support Policy P” and I disagree, I would wonder whether that person would support policy P in a world where F was not true, I would wonder whether I would support P in a world where F was true, I would wonder whether we have different evidence for F, etc.
At least like that the argument can focus on something more specific that we actually disagree on, such as “in a hypothetical world where F was true, would policy P be a good thing?” or “Is F actually true?”—or we may even realize that we don’t disagree at all, that we were just using terms a bit differently (For example “Male engineers should get higher pay than female engineers” was being interpreted as “Companies should give extra money to engineers who happen to be male” by one party and “The fact that the average salary of male engineers is higher than the average salaries of female engineers is not in itself evidence of discrimination” by the other).
Bad choice of example, but I think I agree with your main point—it’s a common mistake to focus on one component of an argument when it isn’t your main objection, where that fact being true or false will not change your final conclusion much.
However, I’m not a big fan of the general argumentation/”arguments as soldier” context. In such a situation, I would not try to find the argument that best demolishes the other person’s argument, I would try to work out on what exactly we agree and on what exactly we disagree. For example, if someone says “Fact F is true, therefore we should support Policy P” and I disagree, I would wonder whether that person would support policy P in a world where F was not true, I would wonder whether I would support P in a world where F was true, I would wonder whether we have different evidence for F, etc.
At least like that the argument can focus on something more specific that we actually disagree on, such as “in a hypothetical world where F was true, would policy P be a good thing?” or “Is F actually true?”—or we may even realize that we don’t disagree at all, that we were just using terms a bit differently (For example “Male engineers should get higher pay than female engineers” was being interpreted as “Companies should give extra money to engineers who happen to be male” by one party and “The fact that the average salary of male engineers is higher than the average salaries of female engineers is not in itself evidence of discrimination” by the other).