The scripture from the article is Prov. 26:20 not Prov. 26:30, which in NWT reads: "Where there is no wood, the fire goes out, And where there is no slanderer, quarreling ceases."
I don't see what's so controversial here to rile your relative. I agree with your point that Matthew 5:23,24 is about straightening out problems with others in the congregation, and the org does still use that verse too. But I don't see anything wrong with suggesting people think seriously about Prov 26:20 first.
In other words, first think "Is this really such a big deal, or can I forget about it, to prevent possibly causing an even bigger issue and making things blow out of proportion?" (the Proverbs 26:20 bit - don't fuel the fire and don't be tempted to diss/slander the person you've have an issue with to others).
But if you decide, "no I can't just leave it I have to say something", maybe because the other person is affected by the difference you had, or it's damaging your relationship with them, then Matthew 5:23,24 comes into play and you discuss the issue with the person directly.
I see them as complementary points, not contradictory or opposites. Also, the context of that paragraph in the WT was if we think someone has offended us, not if someone else has something against us, so maybe that's why they didn't use Matthew 5 there.
Of course, the above reasoning should be obvious without needing scriptures to back it up (many people with no Bible knowledge or interest apply the same principles in their life), but those verses do support the basic ideas behind conflict resolution.