Teddy was a century ago, "progressive" for his era but only in some ways (Imperialist for one thing).
The Populists weren't as "progressive" as some later day authors (Kazin, et al) tried to pretend to co-opt them.
They were mostly rural farmers who were objecting to globalism and industrialization, they believed in direct democracy for themselves, were opposed to unwashed immigrants and urban power centers, blamed the corporations for all their economic ills. Once the panic of 1896 was over and agricultural markets rebounded, the better off farmers became conservative and left their poorer ilk in the ditch, supporting things like poll taxes and so forth to disenfranchise poor farmers and farm workers - populist voting was correlated with distance from railroads (i.e. poorer land and higher cost of getting crops to market). And once the two bad decades were over, white Populists abandoned efforts to create a multi-racial coalition.
The original Progressives were an elitist movement (similar to the current one) that took many of the Populist demands and universalized them - they were primarily urban based so they didn't have the xenophobia of rural America, though they did patronize the people they endeavored to help.
Real change is always incremental (because you have to build 60% support to avoid cycling when measures passed by narrow margins are easily reversed after the next election) - even the New Deal failed in many of its programs b/c they were top down (NIRA, etc.) and never developed the base of support that measures like Social Security did. And change often requires unpleasant compromises (Democrats had to deal with the South on its terms to accomplish anything until LBJ finally forced them to change).
The best hope for America is a collapse by the Republicans leading to the election of numerous moderate Democrats from flyover country who can steamroll the Progressives and pass programs that are sustainable and not subject to overreach - like an immigration reform that strengthens the border, limits asylum, speeds up deportration (not just border crossers but airplane visitors who overstay Visas) but grants citizenship to DACAs and any immigrant who serves in the armed forces.