Turns out the GSS has a potentially relevant question from 2002 in which respondents were asked how often they'd been in contact with a cousin in the past four weeks.
Excluding those who did not have any living cousins, I created a simple index of extended family closeness (EFC), by self-reported ethnicity, by giving 2 points for the percentages of respondents who had been in contact with a cousin more than twice in over the last four weeks, 1 point for the percentages who had been in contact once or twice over the same period of time, and no points for the percentages who had no contact with a cousin. Thus the higher the score, the closer the extended family (clannish) ties tend to be. Because the question was only asked in a single iteration of the survey, sample sizes by ethnicity are pretty small. Only responses for ethnicities with at least 25 respondents are included here. The data are suggestive, not statistically significant, so make of them what you will:
Given the small sample sizes and inherent imprecision of self-described ethnicity, these results pass the smell test. Excepting those of Scottish descent--maybe these are all low-landers!--the rank ordering is pretty close to what I would've expected it to be. Additionally, I'd have guessed the English/Welsh and French rankings would be flipped and that "Americans" would have come in between Italians and Irish. Many of those who self-identify as "American" are what we might also refer to as "Scotch-Irish" [edit: Only 39% are white while 52% are black and 3% are Hispanic, so the black element is far more explanatory than the potential Scotch-Irish element here, thanks M], so if we plug them roughly into the Irish and Scottish figures, the table has even more stereotypical validity.
GSS variables used: COUSINS(1-3), ETHNIC(1,8,10,11,14,15,17,24,30,97)