
Anthony C. answered 03/07/21
Enthusiastic and Experienced History and Political Tutor
There are two ways to answer this question- what the president's role is *supposed* to be, and what it is.
Originally the president wasn't supposed to have a large role- at all. The only constitutional provision dealing with the president and the process of lawmaking is really the veto. But in the beginning even that was used sparingly- presidents only vetoed laws they believed were flat-out unconstitutional. When Andrew Jackson vetoed laws that he simply didn't like (such as the re-charter of the Second Bank of the United States), he was called tyrannical by political opponents.
But once Jackson broke that norm, it never really came back. Over time presidents gained more and more power, especially as party organizations became stronger and our "democratic republic" shifted to having more emphasis on "democratic" and less on "republic."
Now, it's pretty substantial. President Trump was unique in that his White House didn't play a massive and clear role in terms of trying to form bills- many recent presidents have directed the political process (at least their White House staffs have).