Fully agreed; this is something that always baffles me when it's misunderstood so often. Regardless of whether it's logical or not, tone and attitude in practice does influence whether people are convinced by something, so if your goal is to actually change how someone else acts, you will not be as effective if you don't care about how you come across. Being right is not always enough, so even if the style of communicating doesn't seem like it "should" matter, in practice it genuinely does if success is measured by whether the change happens or not.

Of course, if the goal is just to be right rather than to convince someone else about what's right, how you're saying something doesn't matter, but at that point you've already reached the goal before you started talking to them, so it's worth reexamining what you're actually looking to get out of a conversation at that point.