What political belief is often linked to late-nineteenth-century US history and stresses the needs and concerns of the common populace, not the elites of society?

I believe you're referring to the political movement known as Progressivism, which was hugely influential in the United States in the late 19th and early 20th centuries. Progressivism emerged out of the so-called Gilded Age, a time of growing wealth and opportunity in America. On the surface, the United States seemed wealthy, confident, and strong, with the highest standard of living of any country in the world. At the same time, there was a seedy underbelly to the system of capitalism responsible for American wealth and power, which the Progressivist movement set out to expose. Progressives highlighted numerous abuses, such as child labor, the exploitation of workers, and the unsanitary conditions in the food manufacturing industry.
In relation to democracy, the Progressivists sought to take power out of the hands of elite political machines and give it instead to the ordinary people, the voters. Party machines such as Tammany Hall in New York were notorious for graft and corruption. Progressivists thought that innovations such as single-issue referenda and the power to recall elected officials were a good way of tackling the entrenched power of political elites and the corruption to which it all too often led. Active citizen participation at every stage of the democratic process was a vital principle for the Progressive movement and its supporters. It was only if the people themselves actively got involved in what was, after all, supposed to be their democracy, that the institutions of government could be properly controlled, monitored, and held accountable.