Thomas offers a plausible account of why the public would be receptive to slogans like "judges should interpret the law, not make it," but I'm not persuaded that the vast majority of voters pay enough attention to questions of jurisprudence even to distinguish between the bumper sticker versions of originalism/textualism on the one hand and living constitutionalism on the other hand. What people care about is results. That's why, as Barry Friedman has argued, criticism aimed at the countermajoritarian character of the Supreme Court has tended to erupt at just those moments when the Court's decisions have been especially unpopular on substantive grounds.
This explains why news coverage of the use of terms like "strict constructionist" tends to emphasize that this is a code phrase for "overturn Roe v. Wade." It is intended as such code, even though one could actually be a strict constructionist or textualist or whateverist on a whole range of issues. Lawyers, party activists on social issues, and some well-educated non-lawyers may understand these differences in ways that go beyond their implications for a handful of particular issues, but the vast majority of Americans simply don't follow the courts closely enough to have a clear view about jurisprudence. (See my May 2002 column on the issue of Americans' understanding of our Constitution here.)
Indeed, on those extremely rare occasions when the public does become engaged in matters of jurisprudence, textualism/originalism doesn't fare very well. I have in mind the Bork confirmation hearings, although admittedly that was two decades ago and Bork didn't play well on tv as a matter of style, so perhaps even that event wasn't about jurisprudence. It is instructive, though, that successful Republican nominees have done well by emphasizing their pragmatism rather than textualism/originalism as such. Roberts is the paradigm here. To be sure, he didn't embrace the living Constitution as such, but nearly all of what he said as a matter of general jurisprudence could have been said by Ginsburg or Stevens, and much of it by Brennan or Marshall.