"They give these important families political support."
Translation:Zij geven deze belangrijke families politieke steun.
"Gezin" is only used to describe the immediate family, parent(s) and child(ren).
"Familie" can refer to "gezin", but also to all related relatives, grandparents, uncles, aunts, cousins etc.
In other words, "mijn gezin" and "mijn familie" both translate to my family in English but they have a different connotation, as with the later you generally refer to all you relatives.
Hypothetically you could say "de belangrijke gezinnen", though in a sentence like this that would be uncommon. For me personally, and I believe for most people, I would hardly ever use the plural "gezinnen" as in general "gezin" is only used to specifically refer to someone's immediate family.
Yes, with the addition of "politieke" before "steun" to stick with the English sentence.