Both the president and the federal government in general had a much smaller role in American life prior to FDR.
Although the president has always had the veto power, it was originally conceived as more of a check on Congress only in rare circumstances (see e.g.
this wiki page). Even after the president started making use of the veto power more frequently starting with Andrew Jackson, they generally weren't seen as the legislative agenda-setter until the Progressive Era at the earliest (both Theodore Roosevelt and Woodrow Wilson played big roles in establishing the president as someone who would set the legislative agenda and, especially in Wilson's case, take charge of shepherding bills through Congress). To be clear, I'm not trying to say that presidents never pursued legislative agendas or impacted legislation prior to TR, but it was certainly a lot different from recent presidents who come into office with comprehensive agendas of bills they try to pass through Congress.
More generally, the federal government often had little impact on the day-to-day lives of average citizens, except in times of war. Prior to the Progressive Era there usually wasn't an income tax (except for a brief period during and after the Civil War), there were relatively few federal organizations (to give a few examples of things that didn't exist: CIA, EPA, FDIC, Federal Reserve), and there certainly weren't federal programs like Social Security or Medicare. The average American might have little contact with the federal government outside of the Post Office. So, except in times of crisis, there was often relatively little for the president to actually do, at least as compared to the ubiquitous problem solver that Americans seem to expect today. Although there was at least one federal regulatory agency prior to TR, his presidency really increased the role of the federal government (and the president in particular) in economic life with the establishment of the Food and Drug Agency, the Hepburn Act, various conservationist policies, and anti-trust prosecutions.
Finally, after Jackson, there was a decline in the power of individual presidents in favor of the power of parties, as perhaps best evidenced by the fact that 5 of the first 7 presidents won two terms, but 3 of the next 16 presidents won two terms. There's more to say here, but this answer is probably long enough already.
