I would have thought the reasons for approaching family members is obvious: that their embarrassment at being publicly named as close relatives will cause them to pressure the gett refuser to stop his vindictive and cruel behavior and give the gett.
And as a sidebar: if I still had children to marry off, I’d insist on a Halachic pre-nup, which protects both parties. I will push for it for my grandchildren, 3 of whom are rapidly approaching shidduchim.