The president's Job is to be commander in chief of the miltary. The non-constitution aspect of the job of the President is to use the bully pulpit.
The bully pulpit basically means being in a position of persuading the public. Lots of people have this "power". People seriously running for president are in this position. The Media actually controls this and has various "talking heads" who are constantly using this power. Celebrities have this power. Talkshow hosts like say Oprah, or whoever.
The president generally uses the bully pulpit to make Congress do things, by focusing public attention on what the members do or do not do. It's not really pressure from the President, it's the President getting the public to put presssure on Congress. The president does have the power of veto, but this veto can be overridden and he is the top executive of federal govt. With a larger Federal govt the President does have more power, but the Congress is ultimately in control because they pass the legislation which makes money available for anything in any part of the Federal govt- they have the power of the purse.