The president effectively does because of Veto. They inly cant control the most bipartisan aspects which is very limited today.

The president has a huge say in which bills and budget items are passed, they routinely put enormous pressure on individual senators to whip votes for their agenda. This goes double or even triple for Trump! Congress might literally create the budget, but in our age they're – at the bare minimum – starting from the president's agenda and working with him to pass what he wants first.