At the risk of sounding like one of those people you warn about in this article, I would like to make a few comments. (By the way I have taught history for over twenty years in a blue state and am very concerned about the direction of the Republican party.)
You are correct that the United States attempted to become more democratic during the progressive era. But it was a strange mishmash. Women got the right to vote, we had the direct election of senators, but also an effort to ban alcohol by rural voters who knew their power was waning.
Many of Wilson's new organizations such as the Federal Trade Commission and the Federal Reserve were designed to solve problems administratively, but ran afoul of traditional constitutional interpretation. The New Deal had the same problem, until the so called "Constitutional Revolution of 1937.
Democracy did not remove all ideological challengers by 2000, because China had crushed dissent in 1989 and the Communist Party maintained one party control. Furthermore movement conservatives had been organizing to overturn liberal decisions on the court since the 1930s and with renewed fervor beginning in the late 1970s.
Our Constitutional structure is still designed to "control the passions of the mob" as our Federalist Papers friends might remind us. The challenge as I see is it we need a government that follows popular opinion but also is informed by expertise. Science policy, economic theory, foreign policy, etc are very complicated and often misunderstood by many people. Our two party system also prevents us from considering the full range of perspectives.
Mike Lee may mean his tweets in the way you claim, but he is also reflective his ideological block. My question is how you would go about reforming our constitutional structure to make it more democratic.