That's nonsense. The propagandists take as their one piece of evidence a plank in the Progressive Party platform of 1912: "The protection of home life against the hazards of sickness, irregular employment and old age through the adoption of a system of social insurance adapted to American use."The tip-off as to intention lies in those last four words, "adapted to American use." Germany's "Iron Chancellor," Otto von Bismark, had created in the 1880s insurance provisions for health, accidents, disability, and old age. The health service offered was local, not centralized, with employers contributing one-third of the cost and workers contributing two-thirds. The vague Progressive Party proposal was for something along those lines, and it's hard to know more about Teddy Roosevelt's thinking because this was not one of the issues he emphasized.
If Obama wanted to report accurately Roosevelt's beliefs, he would start with a speech about one of TR's favorite topics, "Deadening socialism."
The 26th president saw governmental control of the economy as the political manifestation of covetousness. He argued that "the only permanently beneficial way in which to help any one is to help him help himself; if either private charity, or governmental action, or any form of social expression destroys the individual's power of self-help, the gravest possible wrong is really done to the individual."
TR also contended that socialism could be fought most successfully by applying biblical ideas about helping the poor. The greatest hope lay through "voluntary action by individuals in the form of associations," particularly when the goal was "that most important of all forms of betterment, moral betterment—the moral betterment which usually brings material betterment in its train." Those who truly wanted to help had to stand "against mere sentimentality, against the philanthropy and charity which are not merely insufficient but harmful."