The paper is based on an impressive data-collection effort over many years. When asked on The Daily Show if that effort was an attempt to "statistically quantify the feeling that everyone has" about money and politics, the authors agreed. Dani Rodrik's recent article suggests that this work has staying power in public discourse, as it comes months after Gilens and Page appeared on television and long after their article received the explainer treatment at Vox ("The new study about oligarchy that's blowing up the Internet, explained"). Given all the attention, I'm puzzled that nobody seems to talk about one issue in the paper.
Not every finding in the article is actually in line with "the feeling that everyone has." Gilens and Page argue that average citizens and mass-based interest groups (as opposed to business interest groups) have little influence. Appendix 1 lists the specific groups that were coded as "mass-based," and there you will find the NRA, AIPAC, and the Christian Coalition in the relatively short list of 11. If you're Jon Stewart, who has previously denounced the pernicious influence of these groups, why don't you mention to the authors when they're on your show that you are surprised or relieved that the NRA and AIPAC have little or no impact?
In fact, I can't recall any media coverage of the study that also mentioned its implicit suggestion that the NRA and other right-leaning groups may not be as powerful as previously thought. It didn't help that the authors used only the AARP and AFL-CIO as examples of mass-based groups as opposed to business groups when on The Daily Show (these are the first two groups in alphabetical order, so no malicious intent, I'm sure). Stewart's immediate reaction was, "so, labor is underrepresented," and the authors confirmed this interpretation of the mass-based finding.
And, though Vox admirably tried to walk readers through the findings of the article, the reporter did not cover those results which distinguished between types of interest groups. He only showed the regression result which suggests that interest groups all lumped together have depressingly more influence on policy than do average citizens, then moved on to link the reader to other depressing articles on the subject. This is an interesting oversight; you don't usually see surprising findings ignored by press coverage.
Note: Originally, there were two questions/observations in this post, but I've retracted the second one until I'm able to think about it some more.