Well, he does, but not with the degree of authority people seem to think he does. The president's will still has a great deal of influence over it, but he's not the entity who has the final say.

However, I noticed Trump himself seems to have been unaware of the full scope of checks and balances that are in place, especially those first few weeks - notably the immigration ban. It's not entirely surprising to me that people don't know about them either. Of course this is anecdote, but I don't recall much if any discussion by the media in the past about who actually approves the budget, just who proposes it. I.E. news outlets focusing on Obama's proposed budget as if it's set in stone, and it seems like they're doing the same with Trump's budget - getting target fixated on what if, not what is.

It's just a theory, but it would make sense to me why so many people seem to be ignorant of the process when many people still rely on the big news agencies for their knowledge of politics and government affairs.