To use an example from the article, if I were to say "Let people enjoy things", and you were to denounce that as a TTC without consideration of my true intent.
In that case you may inadvertently be the one that shuts down the debate prematurely, and I may have actually had a valid perspective.
Essentially if you dismiss someone's argument as false just because it may have had a fallacy within it, that reasoning is itself a fallacy.
Some fallacy-seeking people ironically ignore this and just dismiss anything when they have the "gotcha, you made a fallacy therefore everything you said can be concluded as false" moment.
The idea that a single fallacy in a complex chain of reasoning renders the entire chain invalid just makes sense to programmers, who spend most of their time thinking in a step-wise and linear fashion where each output is the input to something else. This sort of "programmer style" thinking has crept into some surprising parts of society these days.
In real life, things are often more nuanced.
The opposite might be true as well, maybe the example you used can be a way to start a conversation and not end one.
So I've taken more of your kind of approach. Either both people are willing to understand each other at that time or are not - and heck you may be wrong in thinking the other person wants to engage / thinking they don't want to engage and make the wrong call. But in a controller-feedback system, you'll inevitably be wrong sometimes - the point is to course correct.
But then you even run into edge cases with this - sometimes people want to engage on a physical level, but not really engage and try to understand. But then how do you differentiate that from the possibility that you see that in others unfairly?
Ah, life is complicated.
In effect, you end up getting them to agree with some otherwise unthinkable positions, just one plateau at a time. There's only so much erosion that can take place before they fall back to their own lines of thought termination (like "all I care about is immigration", or whatever). So you end up with a kind of "anchor" that we can both agree on (ex: corporations are fucking us), which still has the hard edge of politics. At that point, all it takes is for the politics to do enough that the hard edges start to erode. But, there's no accounting for that. Just gotta assume the people who are doing wrong will keep proving it (as they historically have been unable to avoid, no matter how hard they try or how long they are successful at it prior).
As you say: life is complicated. I know people will roll their eyes at this answer as much as any other response I could give you. And I know that a political-focused answer isn't directly analogous to many other situations. But, my answer is as simple as I can think to make it. Just meet people where they're meeting you and don't worry about forcing a point.
“Just go forward in all your beliefs and prove to me that I am not mistaken in mine.” ― William Hartnell, Doctor Who
I have a similar concept, which is roughly described as "don't extend your own arm onto the chopping block, but make use of other people doing the same". Don't make presumptions, and you will never be wrong. Don't dig holes for yourself. Be kind, respectful, genuine. Don't throw the first punch, but don't let innocent people get hurt. Find common ground. When they extend out their arm, take their hand and guide them towards truth and goodness.