Representative Samples, Public Opinion, and Democratic Legitimacy

To what extent is the “crisis of liberal constitutionalism” linked to a growing disillusionment with the “representation” ostensibly provided by “representative democracy”? Consider the rising interest in alternatives that focus on versions of sortition, ie, the use of lotteries rather than elections to select at least some political leaders. One sees this most interestingly, I believe, in the work of James Fishkin, but there's also the very interesting book Against Elections. Both, I would argue, rely on the perception of most social scientists that a well designed random sample is far more likely to be “representative” of public opinion than the result of an election process. But, one assumes, most laypersons do not share the perspective of the social scientist. Does this create insurmountable problems for constitutional reformers or designers who themselves have become skeptical about election-based theories of legitimate government?