Earlier quoted context omitted.
I know this is treading off-topic, but interestingly, "lefty wacko uber-progressive" Seattle (I put that in quotes as it's not my thoughts, but some of the commenters found on The Seattle Times would likely describe it as such) has the highest percentage of nuclear families in the 50 largest cities in the U.S. Source: http://blogs.seattletimes.com/fyi-guy/2014/12/28/end-of-the-...
I think you and your parent comment may be using two different definitions of "nuclear" though. One definition is the traditional mom-dad-2.3 kids arrangement, but a broader and equally valid definition can include polygamy, gay parents with a surrogate, lesbian parents with a sperm donor, couples where the mom and dad have split and started new families but still have kids in common, and a million other combinations…
Why the nuclear rather than the extended family would be considered important for social stability is anyone's guess. We live in an incredibly atomized society that values personal freedom and autonomy over interdependence and social connection, but that doesn't mean "stability".